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Course: Algebra 2 > Unit 6
Evaluating fractional exponents.
- Evaluating fractional exponents: negative unit-fraction
- Evaluating fractional exponents: fractional base
- Evaluating quotient of fractional exponents
- Evaluating mixed radicals and exponents
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Video transcript
Fractional Exponents
If an exponent of a number is a fraction, it is called a fractional exponent . Exponents show the number of times a number is replicated in multiplication. For example, 4 2 = 4×4 = 16. Here, exponent 2 is a whole number. In the number, say x 1/y , x is the base and 1/y is the fractional exponent.
In this article, we will discuss the concept of fractional exponents, and their rules, and learn how to solve them. We shall also explore negative fractional exponents and solve various examples for a better understanding of the concept.
What are Fractional Exponents?
Fractional exponents are ways to represent powers and roots together. In any general exponential expression of the form a b , a is the base and b is the exponent . When b is given in the fractional form, it is known as a fractional exponent. A few examples of fractional exponents are 2 1/2 , 3 2/3 , etc. The general form of a fractional exponent is x m/n , where x is the base and m/n is the exponent.
Look at the figure given below to understand how fractional exponents are represented.
Some examples of fractional exponents that are widely used are given below:
Fractional Exponents Rules
There are certain rules to be followed that help us to multiply or divide numbers with fractional exponents easily. Many people are familiar with whole-number exponents, but when it comes to fractional exponents, they end up doing mistakes that can be avoided if we follow these rules of fractional exponents .
- Rule 1: a 1/m × a 1/n = a (1/m + 1/n)
- Rule 2: a 1/m ÷ a 1/n = a (1/m - 1/n)
- Rule 3: a 1/m × b 1/m = (ab) 1/m
- Rule 4: a 1/m ÷ b 1/m = (a÷b) 1/m
- Rule 5: a -m/n = (1/a) m/n
These rules are very helpful while simplifying fractional exponents. Let us now learn how to simplify fractional exponents.
Simplifying Fractional Exponents
Simplifying fractional exponents can be understood in two ways which are multiplication and division. It involves reducing the expression or the exponent to a reduced form that is easy to understand. For example, 9 1/2 can be reduced to 3. Let us understand the simplification of fractional exponents with the help of some examples.
1) Solve 3 √8 = 8 1/3
We know that 8 can be expressed as a cube of 2 which is given as, 8 = 2 3 . Substituting the value of 8 in the given example we get, (2 3 ) 1/3 = 2 since the product of the exponents gives 3×1/3=1. ∴ 3 √8=8 1/3 =2.
2) Simplify (64/125) 2/3
In this example, both the base and the exponent are in fractional form. 64 can be expressed as a cube of 4 and 125 can be expressed as a cube of 5. They are given as, 64=4 3 and 125=5 3 . Substituting their values in the given example we get, (4 3 /5 3 ) 2/3 . 3 is a common power for both the numbers, hence (4 3 /5 3 ) 2/3 can be written as ((4/5) 3 ) 2/3 , which is equal to (4/5) 2 as 3×2/3=2. Now, we have (4/5) 2 , which is equal to 16/25. Therefore, (64/125) 2/3 = 16/25.
Multiply Fractional Exponents With the Same Base
To multiply fractional exponents with the same base, we have to add the exponents and write the sum on the common base. The general rule for multiplying exponents with the same base is a 1/m × a 1/n = a (1/m + 1/n) . For example, to multiply 2 2/3 and 2 3/4 , we have to add the exponents first. So, 2/3 + 3/4 = 17/12. Therefore, 2 2/3 × 2 3/4 = 2 17/12 .
How to Divide Fractional Exponents?
The division of fractional exponents can be classified into two types.
- Division of fractional exponents with different powers but the same bases
- Division of fractional exponents with the same powers but different bases
When we divide fractional exponents with different powers but the same bases, we express it as a 1/m ÷ a 1/n = a (1/m - 1/n) . Here, we have to subtract the powers and write the difference on the common base. For example, 5 3/4 ÷ 5 1/2 = 5 (3/4-1/2) , which is equal to 5 1/4 .
When we divide fractional exponents with the same powers but different bases, we express it as a 1/m ÷ b 1/m = (a÷b) 1/m . Here, we are dividing the bases in the given sequence and writing the common power on it. For example, 9 5/6 ÷ 3 5/6 = (9/3) 5/6 , which is equal to 3 5/6 .
Negative Fractional Exponents
Negative fractional exponents are the same as rational exponents. In this case, along with a fractional exponent, there is a negative sign attached to the power. For example, 2 -1/2 . To solve negative exponents , we have to apply exponents rules that say a -m = 1/a m . It means before simplifying an expression further, the first step is to take the reciprocal of the base to the given power without the negative sign. The general rule for negative fractional exponents is a -m/n = (1/a) m/n .
For example, let us simplify 343 -1/3 . Here the base is 343 and the power is -1/3. The first step is to take the reciprocal of the base, which is 1/343, and remove the negative sign from the power. Now, we have (1/343) 1/3 . As we know that 343 is the third power of 7 as 7 3 = 343, we can re-write the expression as 1/(7 3 ) 1/3 . Since 3 and 1/3 cancel each other, the final answer is 1/7.
Related Articles
- Non-Integer Rational Exponents
- Irrational Exponents
- Exponential Terms
- Negative Exponents
Fractional Exponents Examples
Example 1: Evaluate 18 1/2 ÷ 2 1/2 .
Solution: In this question, fractional exponents are given. The powers are the same but the bases are different. Hence, we can solve this problem as, 18 1/2 ÷ 2 1/2 = (18/2) 1/2 = 9 1/2 = 3. Therefore, 3 is the required answer.
Example 2: Solve the given expression involving the multiplication of terms with fractional exponents.
2 1/2 × 4 1/4 × 8 1/8
Solution: 4 can be expressed as a square of 2, i.e. 4 = 2 2 . So, 4 1/4 can be written as (2 2 ) 1/4 . It is equal to 2 1/2 . Now, 8 can be expressed as a cube of 2, i.e. 8 = 2 3 . So, 8 1/8 can be written as (2 3 ) 1/8 . It is equal to 2 3/8 . Therefore, the given expression can be re-written as,
2 1/2 × 2 1/2 × 2 3/8
Multiplication of fractional exponents with the same base is done by adding the powers and writing the sum on the common base.
⇒ 2 (1/2 + 1/2 + 3/8)
Therefore, 2 1/2 × 4 1/4 × 8 1/8 = 2 11/8 .
Example 3: Evaluate 3 2/3 ÷ 9 1/2
Solution: To solve this, we will reduce 9 1/2 to the simplest form. So, we have
9 1/2 = (3 2 ) 1/2
3 2/3 ÷ 9 1/2 = 3 2/3 ÷ 3 1
= 3 2/3 - 1
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Fractional Exponents Questions
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FAQs on Fractional Exponents
What do fractional exponents mean.
Fractional exponents mean the power of a number is in terms of fraction rather than an integer . For example, in a m/n the base is 'a' and the power is m/n which is a fraction .
What is the Rule for Fractional Exponents?
In the case of fractional exponents, the numerator is the power and the denominator is the root. This is the general rule of fractional exponents. We can write x m/n as n √(x m ).
What To Do With Negative Fractional Exponents?
If the exponent is given in negative, it means we have to take the reciprocal of the base and remove the negative sign from the power. For example, 2 -1/2 = (1/2) 1/2 .
How To Solve Fractional Exponents?
To solve fractional exponents, we use the laws of exponents or the exponent rules . The fractional exponents' rules are stated below:
How To Add Fractional Exponents?
There is no rule for the addition of fractional exponents. We can add them only by simplifying the powers, if possible. For example, 9 1/2 + 125 1/3 = 3 + 5 = 8.
How To Divide Fractional Exponents?
Division of fractional exponents with the same base and different powers is done by subtracting the powers, and the division with different bases and same powers is done by dividing the bases first and writing the common power on the answer.
Fractional Exponents
Also called "Radicals" or "Rational Exponents"
Whole Number Exponents
First, let us look at whole number exponents :
The exponent of a number says how many times to use the number in a multiplication.
In this example: 8 2 = 8 × 8 = 64
Another example: 5 3 = 5 × 5 × 5 = 125
But what if the exponent is a fraction?
Let's see why in an example.
First, the Laws of Exponents tell us how to handle exponents when we multiply:
Example: x 2 x 3 = (xx)(xxx) = xxxxx = x 5
Which shows that x 2 x 3 = x (2+3) = x 5
So let us try that with fractional exponents:
Example: What is 9 ½ × 9 ½ ?
9 ½ × 9 ½ = 9 (½+½) = 9 (1) = 9
So 9 ½ times itself gives 9.
Now, what do we call a number that, when multiplied by itself, gives another number? The square root of that other number!
√9 × √9 = 9
9 ½ × 9 ½ = 9
So 9 ½ is the same as √9
Try Another Fraction
Let us try that again, but with an exponent of one-quarter (1/4):
16 ¼ × 16 ¼ × 16 ¼ × 16 ¼ = 16 (¼+¼+¼+¼) = 16 (1) = 16
So 16 ¼ used 4 times in a multiplication gives 16,
and so 16 ¼ is a 4th root of 16
General Rule
It worked for ½ , it worked with ¼ , in fact it works generally:
x 1/ n = The n- th Root of x
In other words:
x 1/n = n √ x
Example: What is 27 1/3 ?
Answer: 27 1/3 = 3 √ 27 = 3
What About More Complicated Fractions?
What about a fractional exponent like 4 3/2 ?
That is really saying to do a cube (3) and a square root (1/2), in any order.
Let me explain.
A fraction (like m/n ) can be broken into two parts:
- a whole number part ( m ) , and
- a fraction ( 1/n ) part
So, because m/n = m × (1/n) we can do this:
x m/n = x (m × 1/n) = (x m ) 1/n = n √ x m
The order does not matter, so it also works for m/n = (1/n) × m :
x m/n = x (1/n × m) = (x 1/n ) m = ( n √ x ) m
And we get this:
Some examples:
Example: What is 4 3/2 ?
4 3/2 = 4 3×(1/2) = √(4 3 ) = √(4×4×4) = √(64) = 8
4 3/2 = 4 (1/2)×3 = (√4) 3 = (2) 3 = 8
Either way gets the same result.
Example: What is 27 4/3 ?
27 4/3 = 27 4×(1/3) = 3 √ 27 4 = 3 √ 531441 = 81
27 4/3 = 27 (1/3)×4 = ( 3 √ 27 ) 4 = (3) 4 = 81
It was certainly easier the 2nd way!
Now ... Play With The Graph!
See how smoothly the curve changes when you play with the fractions in this animation, this shows you that this idea of fractional exponents fits together nicely:
Things to try:
- Start with m=1 and n=1, then slowly increase n so that you can see 1/2, 1/3 and 1/4
- Then try m=2 and slide n up and down to see fractions like 2/3 etc
- Now try to make the exponent −1
- Lastly try increasing m, then reducing n, then reducing m, then increasing n: the curve should go around and around
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1.5: Equations with Rational Exponents
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We have solved linear equations, rational equations, radical equations, and quadratic equations using several methods. However, there are many other types of equations, such as equations involving rational exponents, polynomial equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules.
Solving Equations Involving Rational Exponents
Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, \({16}^{\tfrac{1}{2}}\) is another way of writing \(\sqrt{16}\); \(8^{\tfrac{1}{3}}\) is another way of writing \(\sqrt[3]{8}\). The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.
Equations in which a variable expression is raised to a rational exponent can be solved by raising both sides of the equation to the reciprocal of the exponent. The reason the expression is raised to the reciprocal of its exponent is because the product of a number and its reciprocal is one. Therefore the exponent on the variable expression becomes one and is thus eliminated.
Definition: Rational Exponents
A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:
\[a^{\tfrac{m}{n}}={\left (a^{\tfrac{1}{n}} \right )}^m={(a^m)}^{\tfrac{1}{n}}=\sqrt[n]{a^m}={(\sqrt[n]{a})}^m \nonumber\]
Example \(\PageIndex{1}\): Evaluate a Number Raised to a Rational Exponent
Evaluate \(8^{\tfrac{2}{3}}\)
Solution. It does not matter whether the root or the power is done first because \(8^{\tfrac{2}{3}} = (8^2)^{\tfrac{1}{3}}= (8^{\tfrac{1}{3}})^2\). Since the cube root of \(8\) is easy to find, \(8^{\tfrac{2}{3}}\) can be evaluated as \({\left (8^{\tfrac{1}{3}} \right )}^2= {(2)}^2= 4\).
Evaluate \({64}^{-\tfrac{1}{3}}\)
\(\dfrac{1}{4}\)
- Isolate the expression with the rational exponent
- If the numerator of the reciprocal power is an even number, the solution must be checked because the solution involves the squaring process which can introduce extraneous roots.
- If the denominator of the reciprocal power is an even number, this is equivalent to taking an even root so +/- must be included.
Example \(\PageIndex{2}\): Solve an Equation Containing a Variable Raised to a Rational Exponent
Solve the equation in which a variable is raised to a rational exponent: \(x^{\tfrac{3}{4}} = 8\).
Solution The exponent on \(x\) is removed by raising both sides of the equation to a power that is the reciprocal of \(\dfrac{3}{4}.\) The reciprocal of \(\dfrac{3}{4}\) is \(\dfrac{4}{3}\). The numerator of this exponent we are applying is an even number, which means that both sides are being raised to an even power.
\[\begin{align*} x^{\tfrac{3}{4}}&= 8\\ {\left(x^{\tfrac{3}{4}}\right)}^{\tfrac{4}{3}}&= {\left(8\right)}^{\tfrac{4}{3}}\\ x&= (8^{1/3})^4\\ &= (2)^4\\ &= 16 \end{align*}\]
It is necessary to check our result because the solution involved raising both sides of the equation to an even power. Raising both sides of an equation to an even power can introduce "extraneous" roots. Therefore our answer must be checked: \( 16 ^ {\tfrac{3}{4}} = (16^ \tfrac{1}{4})^3= 2^3=8 \). \(\color{Cerulean}{✓} \) The solution set is \( \{ 16 \} \).
Example \(\PageIndex{3}\)
Solve \(x^\tfrac{5}{4}+36=4\).
Solution
\(x^\frac{5}{4}=-32\) \( (x^\frac{5}{4})^\frac{4}{5} =(-32)^\frac{4}{5} \) \(x = (\sqrt[5]{-32}) ^ 4 \) \(x = (-2) ^ 4 \) \(x = 16 \)
It is necessary in this case to check our result because the solution involved raising both sides of the equation to an even power. Raising both sides of an equation to an even power can introduce "extraneous" roots. \( 16^\tfrac{5}{4}+36= (\sqrt[4]{16}) ^ 5 +36= 2^5 +36 = 32 + 36 =68 \ne 4 \). Therefore, the solution \(x=16\) must be rejected. Therefore this problem has no solution. The solution set is \( \{ \quad \} \).
Example \(\PageIndex{4}\)
Solve \(x^\tfrac{4}{3}=81\)
Solution . The solution involves raising both sides of the equal sign to the power of \( \frac{3}{4} \). Because the denominator is an even number, that means tha we are actually taking the even root of a quantity, which could be either a positive or negative value.
\( (x^\frac{4}{3})^\frac{3}{4} = { \color{Cerulean}{\pm }}81^\frac{3}{4} \) \(x = \pm (\sqrt[4]{81}) ^ 3 \) \(x = \pm (3) ^ 3 \) \(x = \pm 27 \)
No checking is required in this example because the process did not involve raising both sides of the equation to an even power. The even number was in the denominator, not the numerator of the reciprocal power. The solution set is \( \{ -27, 27 \} \).
Example \(\PageIndex{5}\)
Solve \((x+5)^\tfrac{2}{3}= 64\)
Solution. Notice here that the reciprocal power has an even denominator which represents taking the square root of both sides of the equation. This requires using \( \pm \) in the solution process.
\( ((x+5)^\frac{2}{3})^\frac{3}{2} = \pm(64^\frac{3}{2}) \) \(x+5 =\pm (\sqrt{64}) ^ 3 \) \(x+5 = \pm (8) ^ 3 \) \(x+5 = \pm 512 \) \(x = -5+512 \) and \(x = -5-512 \) \(x = 509 \) and \(x = -517 \)
The solution does not need to be checked! Solution Set: \( \{ 509, -517 \} \)
Solve the equation
a. \( \{ 129, -121 \} \qquad \) b. \( \{ -1 \} \qquad \) c. \( \{ \} \)
Example \(\PageIndex{7}\): Solve an Equation involving Rational Exponents and Factoring
Solve \(3x^{\tfrac{3}{4}} = x^{\tfrac{1}{2}}\).
This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.
\[\begin{align*} 3x^{\tfrac{3}{4}}-\left(x^{\tfrac{1}{2}}\right)&= x^{\tfrac{1}{2}}-\left(x^{\tfrac{1}{2}}\right)\\ 3x^{\tfrac{3}{4}}-x^{\tfrac{1}{2}}&= 0 \end{align*}\]
Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. The factor with the lowest exponent is \(x^{1/2}\), so \(x^{3/4}\) needs to be rewritten as a product involving \(x^{1/2}\).
\[\begin{align*} 3x^{\tfrac{3}{4}}-x^{\tfrac{1}{2}}&= 0\\ 3x^{(\tfrac{1}{2}+\tfrac{1}{4})}-x^{\tfrac{1}{2}}&= 0\\ 3x^{\tfrac{1}{2}}x^{\tfrac{1}{4}}-x^{\tfrac{1}{2}}&= 0\\ x^{\tfrac{1}{2}}\left (3x^{\tfrac{1}{4}}-1 \right )&= 0 \end{align*}\]
Now we have two factors and can use the zero factor theorem.
\( x^{\tfrac{1}{2}}\left (3x^{\tfrac{1}{4}}-1 \right )= 0 \)
\( \begin{array}{c|rl} x^{\tfrac{1}{2}}= 0 \qquad & 3x^{\tfrac{1}{4}}-1&= 0\\ x= 0 \qquad & 3x^{\tfrac{1}{4}}&= 1\\ & x^{\tfrac{1}{4}}&= \dfrac{1}{3} \\ & {\left (x^{\tfrac{1}{4}} \right )}^4&= {\left (\dfrac{1}{3} \right )}^4 \\ & x&= \dfrac{1}{81} \end{array} \)
The solution set is \( {\Large\{} 0, \dfrac{1}{81} {\Large\}}\).
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- Fractional Exponents – Explanation & Examples
What is a Fractional Exponent?
How to multiply fractional exponents with the same base, how to divide fractional exponents, negative fractional exponents, practice questions, fractional exponents – explanation & examples.
The rules for solving fractional exponents become a daunting challenge to many students. They will waste their valuable time trying to understand fractional exponents but, this is of course a huge mishmash in their minds. Don’t worry. This article has sorted out what you need to do in order to understand and solve problems involving fractional exponents
The first step to understanding how to solve fractional exponents is getting a quick recap what exactly they are, and how to treat the exponents when they are combined either by dividing or multiplication.
A fractional exponent is a technique for expressing powers and roots together. The general form of a fractional exponent is:
b n/m = ( m √ b ) n = m √ (b n ), let us define some the terms of this expression.
The radicand is the under the radical sign √. In this case our radicand is b n
- Order/Index of the radical
The index or order of the radical is the number indicating the root being taken. In the expression: b n/m = ( m √ b ) n = m √ (b n ), the order or index of radical is the number m.
This is the number whose root is being calculated. The base is denoted with a letter b.
How to Solve Fractional Exponents?
Let’s know how to solve fractional exponents with the help of examples below.
- Calculate: 9 ½ = √9
= (3 2 ) 1/2
- Solve: 2 3/2 = √ (2 3 )
- Find: 4 3/2
4 3/2 = 4 3× (1/2)
= √ (4 3 ) = √ (4×4×4)
= √ (64) = 8
Alternatively;
4 3/2 = 4 (1/2) × 3
= (√4) 3 = (2) 3 =
- Find the value of 27 4/3 .
27 4/3 = 27 4 × (1/3)
= ∛ (27 4 ) = 3 √ (531441) = 81
27 4/3 = 27 (1/3) × 4
= ∛ (27) 4 = (3) 4 = 81
- Simplify: 125 1/3 125 1/3 = ∛125 = [(5) 3 ] 1/3 = (5) 1 = 5
- Calculate: (8/27) 4/3 (8/27) 4/3 8 = 2 3 and 27 = 3 3 So, (8/27) 4/3 = (2 3 /3 3 ) 4/3 = [(2/3) 3 ] 4/3 = (2/3) 4 = 2/3 × 2/3 × 2/3 × 2/3 = 16/81
Multiplying terms having the same base and with fractional exponents is equal to adding together the exponents. For example:
x 1/3 × x 1/3 × x 1/3 = x (1/3 + 1/3 + 1/3)
= x 1 = x
Since x 1/3 implies “the cube root of x ,” it shows that if x is multiplied 3 times, the product is x.
Consider another case where;
x 1/3 × x 1/3 = x (1/3 + 1/3)
= x 2/3 , this can be expressed as ∛x 2
Workout: 8 1/3 x 8 1/3
8 1/3 x 8 1/3 = 8 1/3 + 1/3 = 8 2/3
And since the cube root of 8 can be found easily,
Therefore, ∛8 2 = 2 2 = 4
You may also come across multiplication of fractional exponents having different numbers in their denominators, in this case, the exponents are added the same way fractions are added.
x 1/4 × x 1/2 = x (1/4 + 1/2)
= x (1/4 + 2/4)
When dividing fractional exponent with the same base, we subtract the exponents. For instance:
x 1/2 ÷ x 1/2 = x (1/2 – 1/2)
= x 0 = 1
This implies that, any number divided by itself is equivalent to one, and this makes sense with the zero-exponent rule that, any number raised to an exponent of 0 is equals one.
16 1/2 ÷ 16 1/4 = 16 (1/2 – 1/4)
= 16 (2/4 – 1/4)
You can notice that, 16 1/2 = 4 and 16 1/4 = 2.
If n/m is a positive fractional number and x > 0; Then x -n/m = 1/x n/m = (1/x) n/m , and this implies that, x -n/m is the reciprocal of x n/m .
In general; if the base x = a/b,
Then, (a/b) -n/m = (b/a) n/m .
Calculate: 9 -1/2
Solve: (27/125) -4/3
Solution (27/125) -4/3 = (125/27) 4/3 = (5 3 /3 3 ) 4/3 = [(5/3) 3 ] 4/3 = (5/3) 4 = (5 × 5 × 5 × 5)/ (3 × 3 × 3 × 3) = 625/81
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Dealing with fractional exponents
How fractional exponents are related to roots
In this lesson we’ll work with both positive and negative fractional exponents. Remember that when ???a??? is a positive real number, both of these equations are true:
???x^{-a}=\frac{1}{x^a}???
???\frac{1}{x^{-a}} = x^a???
The rule for fractional exponents:
When you have a fractional exponent, the numerator is the power and the denominator is the root. In the variable example ???x^{\frac{a}{b}}???, where ???a??? and ???b??? are positive real numbers and ???x??? is a real number, ???a??? is the power and ???b??? is the root.
???x^{\frac{a}{b}}??? ???=??? ???\sqrt[b]{x^a}???
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Changing fractional exponents into roots and vice versa
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Want to learn more about algebra 2 i have a step-by-step course for that. :), an example where we change the fractional exponent into a root in order to simplify the expression.
Simplify the expression.
???4^{\frac{3}{2}}???
In the fractional exponent, ???3??? is the power and ???2??? is the root, which means we can rewrite the expression as
???\sqrt{4^3}???
???\sqrt{4 \cdot 4 \cdot 4}???
???\sqrt{64}???
in a fractional exponent, think of the numerator as an exponent, and the denominator as the root
Another rule for fractional exponents:
To make a problem easier to solve you can break up the exponents by rewriting them. For example, you can write ???x^{\frac{a}{b}}??? as
???\left[(x)^a\right]^{\frac{1}{b}}???
???\left[(x)^{\frac{1}{b}}\right]^a???
Let’s do a few examples.
Let's do another example.
???\left(\frac{1}{9}\right)^{\frac{3}{2}}???
???9??? is a perfect square so it can simplify the problem to find the square root first. We can rewrite the expression by breaking up the exponent.
???\left[\left(\frac{1}{9}\right)^{\frac{1}{2}}\right]^3???
Raising a value to the power ???1/2??? is the same as taking the square root of that value, so we get
???\left[\sqrt{\frac{1}{9}}\right]^3???
???\left(\frac{\sqrt{1}}{\sqrt{9}}\right)^3???
???\left(\frac{1}{3}\right)^3???
This is the same as
???\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)\left(\frac{1}{3}\right)???
???\frac{1}{27}???
How to get rid of fractional exponents
Write the expression without fractional exponents.
???\left(\frac{1}{6}\right)^{\frac{3}{2}}???
We can rewrite the expression by breaking up the exponent.
???\left[\left(\frac{1}{6}\right)^3\right]^{\frac{1}{2}}???
???\left(\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}\right)^{\frac{1}{2}}???
???\sqrt{\frac{1}{6} \cdot \frac{1}{6} \cdot \frac{1}{6}}???
???\sqrt{\frac{1}{216}}???
???\frac{\sqrt{1}}{\sqrt{216}}???
???\frac{1}{\sqrt{36 \cdot 6}}???
???\frac{1}{\sqrt{36} \sqrt{6}}???
???\frac{1}{6\sqrt{6}}???
We need to rationalize the denominator.
???\frac{1}{6\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}???
???\frac{\sqrt{6}}{6 \cdot 6}???
???\frac{\sqrt{6}}{36}???
What happens if you have a negative fractional exponent?
You should deal with the negative sign first, then use the rule for the fractional exponent.
???4^{-\frac{2}{5}}???
First, we’ll deal with the negative exponent. Remember that when ???a??? is a positive real number, both of these equations are true:
???\frac{1}{4^{\frac{2}{5}}}???
In the fractional exponent, ???2??? is the power and ???5??? is the root, which means we can rewrite the expression as
???\frac{1}{\sqrt[5]{4^2}}???
???\frac{1}{\sqrt[5]{16}}???
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Steps for Solving Equations with Fractional Exponents: 1. isolate the variable that has a fractional exponent. 2. convert from a fractional exponent to a
Learn how to deal with Rational Powers or Exponents. Exponents are shorthand for repeated multiplication of the same thing by itself.
Learn how to solve equations that involve fractional exponents. Also learn about base and exponent values. Easy Step-by-Step Explanation by
solve equations with fractional exponents. ... Fractional exponents with numerators other than 1 | Algebra I | Khan Academy. Khan Academy.
How would you simplify the following: (x^3)^(2/3) My first thought would be to multiply the exponents: 3/1 * 2/3 which would leave me with an exponent of 2.
When we divide fractional exponents with the same powers but different bases, we express it as a1/m ÷ b1/m = (a÷b)1/m.
The exponent of a number says how many times to use the number in a multiplication. So what does a fractional exponent mean?
Equations in which a variable expression is raised to a rational exponent can be solved by raising both sides of the equation to the reciprocal
For example, 3 x 3 x 3 x 3 can be written in exponential form as 34 where 3 is the base and 4 is the exponent. They are widely used in algebraic problems, and
Write the expression without fractional exponents. ... We can rewrite the expression by breaking up the exponent. ... We need to rationalize the