Algebra II : Radicals as Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Radicals As Exponents

\(\displaystyle f(x)=\frac{(\sqrt[4]{x^{3}})^{\frac{2}{5}}}{x^{6}}\)

Which of the following answer choices best simplifies \(\displaystyle f(x)\)?

Possible Answers:

\(\displaystyle f(x)=\frac{1}{x^{\frac{53}{10}}}\)

\(\displaystyle f(x)=x^{\frac{53}{10}}\)

\(\displaystyle f(x)=\sqrt[53]{x^{10}}\)

\(\displaystyle f(x)=\frac{1}{x^{-\frac{53}{10}}}\)

Correct answer:

\(\displaystyle f(x)=\frac{1}{x^{\frac{53}{10}}}\)

Explanation:

The first step to simplifying a problem like this one is to convert all radicals to fractional exponents. Remember the following relationship:

\(\displaystyle \sqrt[n]{x}=x^{\frac{1}{n}}\)

Also keep in mind your exponent rules, especially this one:

\(\displaystyle (x^{m})^{n}=x^{mn}\)

Now, let's get started on this problem. First, we change that radical expression into something with fractional exponents instead.

\(\displaystyle f(x)=\frac{(\sqrt[4]{x^{3}})^{\frac{2}{5}}}{x^{6}}\)

\(\displaystyle f(x)=\frac{(x^{\frac{3}{4}})^{\frac{2}{5}}}{x^{6}}\)

Now we use our exponent rules to simplify the numerator.

\(\displaystyle f(x)=\frac{x^{(\frac{3}{4}\cdot\frac{2}{5})}}{x^{6}}=\frac{x^\frac{3}{10}}{x^{6}}\)

Finally, we simplify the entire fraction:

\(\displaystyle f(x)=\frac{x^{\frac{3}{10}}}{x^{\frac{60}{10}}}=x^{\frac{3}{10}}-x^{\frac{60}{10}}=x^{-\frac{53}{10}}\)

We can leave it like this, but it would be better to write it this way, without negative exponents:

\(\displaystyle f(x)=\frac{1}{x^{\frac{53}{10}}}\)

 

Example Question #1 : Radicals As Exponents

Simplify:

\(\displaystyle \sqrt{\sqrt[3]{4x^{2}-4x-1}}\)

You may assume that the radicand is nonnegative.

Possible Answers:

\(\displaystyle \sqrt {2x+1}\)

\(\displaystyle \sqrt {2x-1}\)

\(\displaystyle \sqrt[3]{2x-1}\)

\(\displaystyle \sqrt[6]{4x^{2}-4x-1}\)

\(\displaystyle \sqrt[3]{2x+1}\)

Correct answer:

\(\displaystyle \sqrt[6]{4x^{2}-4x-1}\)

Explanation:

The polynomial \(\displaystyle 4x^{2}-4x-1\), being a polynomial of degree 2, cannot be a cube of another polynomial. Also, it does not fit the pattern of a perfect square binomial, since its constant term is negative. Therefore, we cannot extract a root of the polynomial to help to simplify it.

We can, however, rewrite each root as a fractional exponent, apply the power of a power property, then convert back, as follows:

\(\displaystyle \sqrt{\sqrt[3]{4x^{2}-4x-1}}\)

\(\displaystyle =\sqrt{\left (4x^{2}-4x-1 \right )^{\frac{1}{3}}}\)

\(\displaystyle =\left ( \left (4x^{2}-4x-1 \right )^{\frac{1}{3}} \right )^{\frac{1}{2}}\)

\(\displaystyle = \left (4x^{2}-4x-1 \right )^{\frac{1}{3} \cdot \frac{1}{2} }\)

\(\displaystyle = \left (4x^{2}-4x-1 \right )^{\frac{1}{6} }\)

\(\displaystyle =\sqrt[6]{ 4x^{2}-4x-1 }\)

Example Question #2 : Radicals As Exponents

Simplify: 

\(\displaystyle \sqrt[3]{\sqrt[5]{x^{16}}}\)

You may assume that \(\displaystyle x\) is a nonnegative real number.

Possible Answers:

\(\displaystyle \sqrt[16]{x ^{15} }\)

\(\displaystyle x \sqrt[15]{x }\)  

\(\displaystyle x^{2}\)

\(\displaystyle \frac{1}{x^{2}}\)

\(\displaystyle x^{8}\)

Correct answer:

\(\displaystyle x \sqrt[15]{x }\)  

Explanation:

The best way to simplify a radical within a radical is to rewrite each root as a fractional exponent, then convert back.

First, rewrite the roots as exponents.

\(\displaystyle \sqrt[3]{\sqrt[5]{x^{16}}} = \sqrt[3]{ \left (x^{16} \right ) ^{\frac{1}{5}} } = \left [ \left (x^{16} \right ) ^{\frac{1}{5}} \right ] ^{\frac{1}{3}} = x^{16 \cdot \frac{1}{5} \cdot \frac{1}{3} } = x ^{\frac{16}{15}} = \sqrt[15]{x^{16}}\)

We can simplify this further:

\(\displaystyle \sqrt[15]{x^{16}} = \sqrt[15]{x^{15}} \cdot \sqrt[15]{x } = x \sqrt[15]{x }\)

Example Question #1 : Radicals As Exponents

What is the value of \(\displaystyle 125^\frac{3}{2}\)?

Possible Answers:

\(\displaystyle 350\sqrt{5}\)

\(\displaystyle 225\sqrt{3}\)

\(\displaystyle 125\sqrt{3}\)

\(\displaystyle 625\)

\(\displaystyle 625\sqrt{5}\)

Correct answer:

\(\displaystyle 625\sqrt{5}\)

Explanation:

Recall that when you have a fractional exponent, this means that you have a root involved.  The denominator of the exponent is the type of root.  Our question's exponent is:

\(\displaystyle \frac{3}{2}\)

Therefore, the root is \(\displaystyle 2\) or a square root.

The numerator is the power for the base.  Therefore, we can rewrite our problem as:

\(\displaystyle 125^\frac{3}{2} = \sqrt{125^3}\)

Now, to simplify this, we could do:

\(\displaystyle \sqrt{125^3}= \sqrt{(5^3)^3}\)

Using our exponent rules, this is:

\(\displaystyle \sqrt{5^9}\)

Factoring out \(\displaystyle 4\) sets of \(\displaystyle 5\), we get:

\(\displaystyle 5^4\sqrt{5}\), or \(\displaystyle 625\sqrt{5}\)

 

Example Question #3 : Radicals As Exponents

What is the value of \(\displaystyle 45^\frac{4}{3}\)?

Possible Answers:

\(\displaystyle 45\sqrt[4]{9}\)

\(\displaystyle 45\sqrt[3]{45}\)

\(\displaystyle 45\sqrt[4]{45}\)

\(\displaystyle 3\sqrt[3]{5}\)

\(\displaystyle 9\sqrt[3]{45}\)

Correct answer:

\(\displaystyle 45\sqrt[3]{45}\)

Explanation:

Recall that when you have a fractional exponent, this means that you have a root involved.  The denominator of the exponent is the type of root.  Our question's exponent is:

\(\displaystyle \frac{4}{3}\)

Therefore, the root is \(\displaystyle 3\) or a cube root.

The numerator is the power for the base.  Therefore, we can rewrite our problem as:

\(\displaystyle 45^\frac{4}{3} = \sqrt[3]{45^4}\)

Now, to simplify this, we could do:

\(\displaystyle \sqrt[3]{45^4} = \sqrt[3]{(3^2*5)^4}\)

Using our exponent rules, this is:

\(\displaystyle \sqrt[3]{3^8*5^4}\)

Factoring out \(\displaystyle 2\) sets of \(\displaystyle 3\) and \(\displaystyle 1\) set of \(\displaystyle 5\), we get:

\(\displaystyle 3^2*5\sqrt[3]{3^2*5}=45\sqrt[3]{45}\)

Example Question #6 : Radicals As Exponents

What is the value of \(\displaystyle 42^\frac{5}{3}\)?

Possible Answers:

\(\displaystyle 14\sqrt[5]{441}\)

\(\displaystyle 21\sqrt[3]{882}\)

\(\displaystyle 14\sqrt[3]{441}\)

\(\displaystyle \sqrt[5]{74088}\)

\(\displaystyle 42\sqrt[3]{1764}\)

Correct answer:

\(\displaystyle 42\sqrt[3]{1764}\)

Explanation:

Recall that when you have a fractional exponent, this means that you have a root involved.  The denominator of the exponent is the type of root.  Our question's exponent is:

\(\displaystyle \frac{5}{3}\)

Therefore, the root is \(\displaystyle 3\) or a cube root.

The numerator is the power for the base.  Therefore, we can rewrite our problem as:

\(\displaystyle 42^\frac{5}{3} = \sqrt[3]{42^5}\)

Now, to simplify this, we could do:

\(\displaystyle \sqrt[3]{42^5} = \sqrt[3]{(2*3*7)^5}\)

We can factor out a set of \(\displaystyle 2*3*7\).  This leaves us with:

\(\displaystyle 2*3*7\sqrt[3]{(2*3*7)^2}\)

Simplifying, this is:

\(\displaystyle 42\sqrt[3]{1764}\)

Example Question #7 : Radicals As Exponents

Which of the following is equivalent to \(\displaystyle \sqrt[3]{2}\)

Possible Answers:

\(\displaystyle 2^{-3}\)

\(\displaystyle 2^{\frac{1}{3}}\)

\(\displaystyle 2^3\)

\(\displaystyle \frac{2}{3}\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 2^{\frac{1}{3}}\)

Explanation:

Recall that the cube root of a number is the number that when multiplied by itself 3 times, yields your number. 

Thus, we want y, where: 

\(\displaystyle y\times y \times y = 2\)

Consider \(\displaystyle 2^{-3} \cdot 2^{-3}\cdot 2^{-3} = 2^{-3 + -3 + -3} = 2^{-9}\)

That cannot be our solution. 

 

Then try \(\displaystyle 2^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}\cdot 2^{\frac{1}{3}} = 2^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}} = 2^{1}\)

Thus, this must be our solution. 

 

Next time, remember that radicals can be represented by fractional exponents! 

Example Question #2 : Radicals As Exponents

Choose the best answer.  Reduce the following in exponential form:  \(\displaystyle (\sqrt{40})^{\frac{1}{2}}\)  

Possible Answers:

\(\displaystyle 2^\frac{1}{4} \times10^\frac{1}{8}\)

\(\displaystyle 2^\frac{1}{2} \times10^\frac{1}{4}\)

\(\displaystyle 20^\frac{1}{4}\)

\(\displaystyle 20^\frac{3}{4}\)

\(\displaystyle 20^\frac{1}{8}\)

Correct answer:

\(\displaystyle 2^\frac{1}{2} \times10^\frac{1}{4}\)

Explanation:

Simplify the inner term within the parentheses.

\(\displaystyle (\sqrt{40})^{\frac{1}{2}} = (\sqrt{4\times 10})^{\frac{1}{2}} =( \sqrt4 \times \sqrt{10})^{\frac{1}{2}}=( 2 \sqrt{10})^{\frac{1}{2}}\)

\(\displaystyle ( 2 \sqrt{10})^{\frac{1}{2}}= 2^\frac{1}{2} \times \sqrt{10}^{\frac{1}{2}}=2^\frac{1}{2} \times ((10)^{\frac{1}{2}})^{\frac{1}{2}}=2^\frac{1}{2} \times10^\frac{1}{4}\)

Example Question #9 : Radicals As Exponents

Simplify,

\(\displaystyle x^3\sqrt{x}\)

Possible Answers:

\(\displaystyle x^{\frac{7}{2}}\)

\(\displaystyle x^{\frac{3}{2}}\)

\(\displaystyle x^6\)

can't be simplified

\(\displaystyle x^5\)

Correct answer:

\(\displaystyle x^{\frac{7}{2}}\)

Explanation:

\(\displaystyle x^3\sqrt{x}\)

 

First write the square root of \(\displaystyle x\) as an exponent, 

\(\displaystyle =x^3x^{1/2}\)

 

From the rules of exponents we know we can simplyfy by adding the exponents, 

\(\displaystyle =x^{3+\frac{1}{2}}\)

\(\displaystyle =x^{\frac{7}{2}}\)

 

 

Example Question #10 : Radicals As Exponents

Simplify:

\(\displaystyle (10x^2z^3)^\frac{2}{3}\)

Possible Answers:

\(\displaystyle xz^2\sqrt[3]{100x^2}\)

\(\displaystyle xz^2\sqrt[3]{100x}\)

\(\displaystyle xz\sqrt[3]{100x}\)

\(\displaystyle 10x^3z^4\sqrt{10x}\)

Correct answer:

\(\displaystyle xz^2\sqrt[3]{100x}\)

Explanation:

To simplify the expression, we must remember that a fraction as a power denotes a radical: the numerator is the power to which the term is taken inside the radical, and the denominator denotes the degree of the root (i.e. 2 means square root, 3 means cube root, etc.)

Rewriting our expression, we get

\(\displaystyle \sqrt[3]{(10x^2z^3)^2}\)

which expanded becomes

\(\displaystyle \sqrt[3]{100x^4z^6}=\sqrt[3]{100\cdot x^3\cdot x\cdot z^6}\)

Now, move the cubes outside of the cube root, after taking their cube root, leaving behind the terms that aren't cubes:

\(\displaystyle xz^2\sqrt[3]{100x}\)

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