Rational Exponents - College Algebra

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Question

Simplify: $25^{\frac{3}{2}}$

Answer

An option to solve this is to split up the fraction. Rewrite the fractional exponent as follows:

$25^{\frac{3}{2}} = (25^{\frac{1}{2}})^3$

A value to its half power is the square root of that value.

$25^{\frac{1}{2}} = \sqrt{25}=5$

Substitute this value back into $(25^{\frac{1}{2}})^3$ .

$(25^{\frac{1}{2}})^3 = (5)^3 = 5\times 5\times 5 = 125$

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Question

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

Answer

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

When dealing with fractional exponents, keep the following in mind: The numerator is making the base bigger, so treat it like a regular exponent. The denominator is making the base smaller, so it must be the root you are taking.

This means that $b^{\frac{4}{5}}$ is equal to the fifth root of b to the fourth. Perhaps a bit confusing, but it means that we will keep , but put the whole thing under $\sqrt[5]{b}$ .

So if we put it together we get:

$\sqrt[5]{b^4}$

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Question

Evaluate $\frac{1}{4}^{-\frac{1}{2}}$

Answer

When dealing with fractional exponents, remember this form:

$\sqrt[a]{x^n}$ is the index of the radical which is also the denominator of the fraction, represents the base of the exponent, and is the power the base is raised to. That value is the numerator of the fraction.

With a negative exponent, we need to remember this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

$\frac{1}{4}^{-\frac{1}{2}}=\frac{1}{\frac{1}{4}^{\frac{1}{2}}}=\frac{1}{\sqrt[2]{\frac{1}{4^1}}}=\frac{1}{\frac{1}{2}}=2$

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Question

$\left ( \frac{125}{343} \right )^{\frac{2}{3}}$

Answer

First, distribute the exponent to both the numerator and denominator of the fraction.

$\frac{125^{\frac{2}{3}}}{343^{\frac{2}{3}}}$

The numerator of a fractional exponent is the power you take the number to and the denominator is the root that you take the number to.

$\frac{\sqrt[3]{125^{2}}}{\sqrt[3]{343^{2}}}$

You can take the cubed root and square the numbers in either order but if you can do the root first that is often easier.

$\frac{5^{2}}{7^{2}}$

$\frac{25}{49}$

This is the answer. Alternatively, you could have squared the numbers first before taking the cubed root.

$\frac{\sqrt[3]{15625}}{\sqrt[3]{117649}}$

$\frac{25}{49}$

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Question

Evaluate:

$\left(\frac{1}{256}\right)^{-\frac{1}{4}}$

Answer

In order to evaluate fractional exponents, we can express them using the following relationship:

$x^{\frac{a}{b}}=\sqrt[b]{x^a}$

In this formula, represents the index of the radical from the denominator of the fraction and is the exponent that raises the base: . When exponents are negative, we can express them using the following relationship:

$x^{-a}=\frac{1}{x^a}$

We can then rewrite and solve the expression in the following way.

$\left(\frac{1}{256}\right)^{-\frac{1}{4}}=\frac{1}{\frac{1}{\sqrt[4]{256}}}=\frac{1}{\frac{1}{4}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{1}{\frac{1}{4}}=1\times \frac{4}{1}=4$

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Question

Evaluate.

$144^{1/2}$

Answer

Exponents raised to a power of <1 can be written as the root of the denominator.

So:

$144^{1/2} = \sqrt[2]{144}=\pm12$

Recall that a square root can give two answers, one positive and one negative.

$\(-12)(-12)=144 \(12)(12)=144$

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Question

Evaluate

$8^{11/5}$

Answer

$8^{11/5}$

The denominator of the exponent "N" is the same as the "N" root of that number.

So

$8^{11/5} = \sqrt[5]{8^{11}} = 2.2$

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Question

Evaluate

$(7^{5/3})^{3/4}$

Answer

This appears more complicated than it is.

$(7^{5/3})^{3/4}$ is really just where

$X=7^{5/3}$

and

Re-written this equation looks like

$(\sqrt[3]{7^5})^{3/4} = 1.667^{3/4}= \sqrt[4]{1.667^3} = 11.386$

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Question

Evaluate

$17^{0.824}$

Answer

$17^{0.824}$ can be seen as , in a scientific calculator use the button where .

$17^{0.824}=10.325$

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Question

Evaluate the given rational exponent:

$8^{\frac{2}{3}}$

Answer

Rational exponents can be simplified by following this common rule:

$a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$

We can apply this concept to the given value in order to evaluate:

$\8^{\frac{2}{3}}=\sqrt[3]{8^{2}}=\sqrt[3]{64}\\sqrt[3]{64}=4$

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Question

Solve for :

$81^{\frac{x}{4}}=27$

Answer

We can use the given property of rational exponents to solve for :

$a^\frac{m}{n}=\sqrt[n]{a^m}$

$\81^{\frac{x}{4}}=\sqrt[4]{81^{x}}=27\27^{4}=81^x\81^{x}=531,441\81^{3}=531,441\x=3$

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Question

Solve for :

$32^{\frac{2}{x}}=4$

Answer

We can use the given property of rational exponents to solve for :

$a^{\frac{m}{n}}=\sqrt[n]{a^m}$

$\32^{\frac{2}{x}}=\sqrt[x]{32^2}=4\\sqrt[x]{1024}=4\4^{x}=1024\4^{5}=1024\x=5$

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Question

Evaluate: $(625)^{\frac {1}{4}}$

Answer

Step 1: We need to understand what the fractional value in the exponent is.

A fractional exponent, $\frac {1}{n}$ , tells us that we must take the th root of the number.

In this case, we have $\frac {1}{4}$ , so we will take the 4th root of .

Step 2: Calculate... $\sqrt_[4]{625}=5$

$\sqrt[4]{625}=\sqrt[4]{555*5}=5$

The answer is .

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