Radicals as Exponents - Algebra II

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Question

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

Which of the following answer choices best simplifies ?

Answer

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

$\sqrt[n]{x}=x^{\frac{1}{n}}$

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

$(x^{m})^{n}=x^{mn}$

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

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

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

Now we use our exponent rules to simplify the numerator.

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

Finally, we simplify the entire fraction:

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

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

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

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Question

Simplify:

$\sqrt{\sqrt[3]{4x^{2}-4x-1}}$

You may assume that the radicand is nonnegative.

Answer

The polynomial $4x^{2}-4x-1$ , 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:

$\sqrt{\sqrt[3]{4x^{2}-4x-1}}$

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

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

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

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

$=\sqrt[6]{ 4x^{2}-4x-1 }$

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Question

Simplify:

$\sqrt[3]{\sqrt[5]{x^{16}}}$

You may assume that is a nonnegative real number.

Answer

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.

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

We can simplify this further:

$\sqrt[15]{x^{16}} = \sqrt[15]{x^{15}} \cdot \sqrt[15]{x } = x \sqrt[15]{x }$

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Question

Which of the following is equivalent to $\sqrt[3]{2}$ ?

Answer

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:

$y\times y \times y = 2$

Consider $2^{-3} \cdot 2^{-3}\cdot 2^{-3} = 2^{-3 + -3 + -3} = 2^{-9}$

That cannot be our solution.

Then try $2^{\frac{1}{3}}\cdot 2^{\frac{1}{3}}\cdot 2^{\frac{1}{3}} = 2^{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}} = 2^{1}$

Thus, this must be our solution.

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

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Question

What is the value of $125^\frac{3}{2}$ ?

Answer

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:

$\frac{3}{2}$

Therefore, the root is or a square root.

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

$125^\frac{3}{2} = \sqrt{125^3}$

Now, to simplify this, we could do:

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

Using our exponent rules, this is:

$\sqrt{5^9}$

Factoring out sets of , we get:

$5^4\sqrt{5}$ , or $625\sqrt{5}$

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Question

What is the value of $45^\frac{4}{3}$ ?

Answer

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:

$\frac{4}{3}$

Therefore, the root is or a cube root.

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

$45^\frac{4}{3} = \sqrt[3]{45^4}$

Now, to simplify this, we could do:

$\sqrt[3]{45^4} = \sqrt[3]{(3^25)^4}$

Using our exponent rules, this is:

$\sqrt[3]{3^85^4}$

Factoring out sets of and set of , we get:

$3^25\sqrt[3]{3^25}=45\sqrt[3]{45}$

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Question

What is the value of $42^\frac{5}{3}$ ?

Answer

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:

$\frac{5}{3}$

Therefore, the root is or a cube root.

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

$42^\frac{5}{3} = \sqrt[3]{42^5}$

Now, to simplify this, we could do:

$\sqrt[3]{42^5} = \sqrt[3]{(237)^5}$

We can factor out a set of . This leaves us with:

$237\sqrt[3]{(237)^2}$

Simplifying, this is:

$42\sqrt[3]{1764}$

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Question

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

Answer

Simplify the inner term within the parentheses.

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

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

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Question

Simplify,

$x^3\sqrt{x}$

Answer

$x^3\sqrt{x}$

First write the square root of as an exponent,

$=x^3x^{1/2}$

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

$=x^{3+\frac{1}{2}}$

$=x^{\frac{7}{2}}$

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Question

Simplify:

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

Answer

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

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

which expanded becomes

$\sqrt[3]{100x^4z^6}=\sqrt[3]{100\cdot x^3\cdot x\cdot z^6}$

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

$xz^2\sqrt[3]{100x}$

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Question

Simplify: $-\frac{1}{16^{\frac{1}{4}}}+81^{\frac{1}{4}}$

Answer

The fractional exponent represents the fourth root of the given number.

Evaluate $16^{\frac{1}{4}}$ .

$16^{\frac{1}{4}}=\sqrt[4]{16} = 2$

Evaluate $81^{\frac{1}{4}}$ .

$81^{\frac{1}{4}}=\sqrt[4]{81} = 3$

Rewrite the fractions.

$-\frac{1}{16^{\frac{1}{4}}}+81^{\frac{1}{4}} = -\frac{1}{2}+3 = 2\frac{1}{2}$

The answer is: $2\frac{1}{2}$

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Question

Rewrite the following expression as a radical: $(10)^{\frac{2}{3}}$

Answer

The denominator of the exponent represents the root and the numerator represents the power of what the radical is raised to.

We can also separate the powers using the product rule of exponents.

Rewrite the expression as follows:

$(10)^{\frac{2}{3}} =(10^2)^{\frac{1}{3}}= 100^{\frac{1}{3}} = \sqrt[3]{100}$

Note that the powers in the product can also be interchanged.

$(10^{\frac{1}{3}})^2= (\sqrt[3]{10})^2$

Either answer provided is correct.

The answer is: $\sqrt[3]{100}$

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Question

Rewrite the radical as an exponent:

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

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent as shown below:

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

In this case, we are done because there are no further simplification steps.

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Question

Rewrite the following radical as an exponent:

$\sqrt[3]{x^{6}y^{2}}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to both terms by multiplying it by the exponents of each term as shown below:

$\sqrt[3]{x^{6}y^{2}}=(x^{6}y^{2})^{\frac{1}{3}}=x^{\frac{6}{3}}y^{\frac{2}{3}}$

From this point simplify the exponents accordingly:

$x^{\frac{6}{3}}y^{\frac{2}{3}}=x^{2}y^{\frac{2}{3}}$

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Question

Rewrite the following radical as an exponent:

$\sqrt[2]{xy^{3}z^{4}}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to all terms by multiplying it by the exponents of each term as shown below:

$\sqrt[2]{xy^{3}z^{4}}=(xy^{3}z^{4})^{\frac{1}{2}}=x^{\frac{1}{2}}y^{\frac{3}{2}}z^{\frac{4}{2}}$

From this point simplify the exponents accordingly:

$x^{\frac{1}{2}}y^{\frac{3}{2}}z^{\frac{4}{2}}=x^{\frac{1}{2}}y^{\frac{3}{2}}z^{2}$

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Question

Rewrite the following radical as an exponent:

$\sqrt[3]{y^{6}z}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to both terms by multiplying it by the exponents of each term as shown below:

$\sqrt[3]{y^{6}z}=(y^{6}z)^{\frac{1}{3}}=y^{\frac{6}{3}}z^{\frac{1}{3}}$

From this point simplify the exponents accordingly:

$y^{\frac{6}{3}}z^{\frac{1}{3}}=y^{2}z^{\frac{1}{3}}$

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Question

Rewrite the following radical as an exponent:

$\sqrt[4]{x^{4}y^{2}}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to all terms by multiplying it by the exponents of each term as shown below:

$\sqrt[4]{x^{4}y^{2}}=(x^{4}y^{2})^{\frac{1}{4}}=x^{\frac{4}{4}}y^{\frac{2}{4}}$

From this point simplify the exponents accordingly:

$x^{\frac{4}{4}}y^{\frac{2}{4}}=xy^{\frac{1}{2}}$

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Question

Rewrite the following radical as an exponent:

$\sqrt[2]{x^{6}y^{5}}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to both terms by multiplying it by the exponents of each term as shown below:

$\sqrt[2]{x^{6}y^{5}}=(x^{6}y^{5})^{\frac{1}{2}}=x^{\frac{6}{2}}y^{\frac{5}{2}}$

From this point simplify the exponents accordingly:

$x^{\frac{6}{2}}y^{\frac{5}{2}}=x^{3}y^{\frac{5}{2}}$

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Question

Rewrite the following radical as an exponent:

$\sqrt[2]{xyz^{2}}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to all terms by multiplying it by the exponents of each term as shown below:

$\sqrt[2]{xyz^{2}}=(xyz^{2})^{\frac{1}{2}}=x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{2}{2}}$

From this point simplify the exponents accordingly:

$x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{2}{2}}=x^{\frac{1}{2}}y^{\frac{1}{2}}z$

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Question

Rewrite the following radical as an exponent:

$\sqrt[3]{x^{2}y^{3}z^{6}}$

Answer

In order to rewrite a radical as an exponent, the number in the radical that indicates the root, gets written as a fractional exponent. Distribute the exponent to all terms by multiplying it by the exponents of each term as shown below:

$\sqrt[3]{x^{2}y^{3}z^{6}}=(x^{2}y^{3}z^{6})^{\frac{1}{3}}=x^{\frac{2}{3}}y^{\frac{3}{3}}z^{\frac{6}{3}}$

From this point simplify the exponents accordingly:

$x^{\frac{2}{3}}y^{\frac{3}{3}}z^{\frac{6}{3}}=x^{\frac{2}{3}}yz^{2}$

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