Understanding Exponents - Algebra II

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Question

Simplify:

$[x^{1/2}]^{7/3}$

Answer

$[x^{m}]^{n}=x^{m\times n}$

$x^{1/2\times 7/3}=x^{7/6}$

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Question

Simplify the expression:

$\small (16^{\frac{1}{2}})(256^{\frac{3}{4}})$

Answer

Remember that fraction exponents are the same as radicals.

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

$256^{\frac{3}{4}}=\sqrt[4]{256^3}=64$

A shortcut would be to express the terms as exponents and look for opportunities to cancel.

$16^{\frac{1}{2}}=(4^2)^{\frac{1}{2}}=4$

$\small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64$

Either method, we then need to multiply to two terms.

$\small (4)(64)=256$

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Question

Convert the exponent to radical notation.

$x^{\frac{3}{7}}$

Answer

Remember that exponents in the denominator refer to the root of the term, while exponents in the numerator can be treated normally.

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

$x^{\frac{3}{7}}=\sqrt[7]{x^3}$

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Question

Write the product of $\small a^\frac{3}{4}a^\frac{3}{8}a^\frac{5}{2}$ in radical form

Answer

This problem relies on the key knowledge that $\small x^\frac{a}{b}=\sqrt[b]{x^a}$ and that the multiplying terms with exponents requires adding the exponents. Therefore, we can rewrite the expression thusly:

$\small a^\frac{3}{4}a^\frac{3}{8}a^\frac{5}{2}=a^{\frac{3}{4}+\frac{3}{8}+\frac{5}{2}}=a^{\frac{29}{8}}=\sqrt[8]{a^{29}}$

Therefore, $\small \sqrt[8]{a^{29}}$ is our final answer.

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Question

Evaluate the following expression:

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}$

Answer

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}} = \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}=\frac{3}{5}$

or

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}= \sqrt[3]{\frac{27}{125}}=\frac{\sqrt[3]{27}}{\sqrt[3]{125}}=\frac{3}{5}$

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Question

Simplify:

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

Answer

Keep in mind that when you are dividing exponents with the same base, you will want to subtract the exponent found in the denominator from the exponent found in the numerator.

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

$\frac{1}{2}-(-\frac{3}{2})=2$

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

Since the exponent is negative, you will want to put the in the denominator in order to make it positive.

So then,

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

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Question

Find the value of $9^{\frac{3}{2}}$ .

Answer

When you have a number or value with a fractional exponent,

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

or

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

So then,

$9^{\frac{3}{2}}=\sqrt[2]{9^3}=\sqrt{729}=27$

$9^{\frac{3}{2}}=(\sqrt9)^3=3^3=27$

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Question

Find the value of $64^{\frac{4}{3}}$

Answer

When you have a number or value with a fractional exponent,

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

or

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

So then,

$64^{\frac{4}{3}}=\sqrt[3]{64^4}=\sqrt[3]{2^{24}}=2^8=256$

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Question

Simplify:

$(27t^3)^\frac{5}{3}$

Answer

When exponents are raised to another exponent, you will need to multiply the exponents together.

When you have a number or value with a fractional exponent,

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

or

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

So,

$27^{\frac{5}{3}}=\sqrt[3]{27^5}=\sqrt[3]{3^{15}}=3^5=243$

$(t^3)^{\frac{5}{3}}=t^{3\times\frac{5}{3}}=t^{\frac{15}{3}}=t^5$

$(27t^3)^\frac{5}{3}=243t^5$

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Question

Simplify:

$\frac{(x^2)(x^{\frac{1}{2}}y^9)}{x^{-\frac{4}{3}}y^5}$

Answer

Start by simplifying the numerator. Since two terms with the same base are being multiplied, add the exponents.

$\frac{(x^2)(x^{\frac{1}{2}}y^9)}{x^{-\frac{4}{3}}y^5}=\frac{x^{\frac{5}{2}}y^9}{x^{-\frac{4}{3}}y^5}$

Now, when terms with the same bases are divided, subtract the exponent from the denominator from the exponent in the numerator.

The exponent for is

$\frac{5}{2}-(-\frac{4}{3})=\frac{23}{6}$

The exponent for is

So then,

$\frac{(x^2)(x^{\frac{1}{2}}y^9)}{x^{-\frac{4}{3}}y^5}=\frac{x^{\frac{5}{2}}y^9}{x^{-\frac{4}{3}}y^5}=x^{\frac{23}{6}}y^4$

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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 $2^{\frac{1}2{}}$

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.

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

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Question

Evaluate $3^{\frac{1}{3}}$

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.

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

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Question

Evaluate $4^{\frac{2}{3}}$

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.

$4^{\frac{2}{3}}=\sqrt[3]{4^2}=\sqrt[3]{16}$

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Question

Evaluate $8^{\frac{1}{3}}$

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.

$8^{\frac{1}{3}}=\sqrt[3]{8^1}=2$

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Question

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

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.

$\frac{1}{2}^{\frac{1}{2}}=\sqrt[2]{\frac{1}{2}^1}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$ To get rid of the radical, just multiply top and bottom by the radical.

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Question

Evaluate $\frac{1}{3}^{\frac{2}{3}}$

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.

$\frac{1}{3}^{\frac{2}{3}}=\sqrt[3]{\frac{1}{3}^2}=\sqrt[3]{\frac{1}{9}}$

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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.

$-\frac{1}{4}^{\frac{1}{2}}=-\sqrt[2]{\frac{1}{4}^1}=-\frac{1}{{2}}$ Remember to keep the negative on the outside. The exponent comes first followed by the negative sign in the end.

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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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