Select the equivalent expression. ((t)/(t^(7)))^((5)/(24)) root(4)(t^(5)) root(5)(t^(4)) (1)/(root(5)(t^(4))) (1)/(root(4)(t^(5)))

Simplify Base: Simplify the base of the given expression. We have ((t)/(t^(7)))^((5)/(24)). First, we simplify the fraction inside the parentheses by using the property of exponents that states when dividing like bases, we subtract the exponents. t^1 / t^7 = t^(1-7) = t^(-6).Apply Exponent: Apply the exponent to the simplified base. Now we have (t^(-6))^((5)/(24)). We use the property of exponents that states (a^m)^n = a^(m*n) to simplify the expression. (t^(-6))^((5)/(24)) = t^((-6)*(5/24)) = t^(-5/4).Convert Negative Exponent: Convert the negative exponent to a positive exponent by taking the reciprocal. t^(-5/4) can be rewritten as 1/(t^(5/4)). This is because a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.Rewrite as Radical: Rewrite the expression as a radical. 1/(t^(5/4)) can be rewritten as a radical expression. The denominator of the exponent (4) becomes the index of the root, and the numerator (5) becomes the exponent inside the root. 1/(t^(5/4)) = 1/root(4)(t^5).Compare with Options: Compare the result with the given options. The expression we have found, 1/root(4)(t^5), matches one of the given options.

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Select the equivalent expression.

((t)/(t^(7)))^((5)/(24))

root(4)(t^(5))

root(5)(t^(4))

(1)/(root(5)(t^(4)))

(1)/(root(4)(t^(5)))

Select the equivalent expression. $\newline$ $\left(\frac{t}{t^{7}}\right)^{\frac{5}{24}}$ $\newline$ $\sqrt[4]{t^{5}}$ $\newline$ $\sqrt[5]{t^{4}}$ $\newline$ $\frac{1}{\sqrt[5]{t^{4}}}$ $\newline$ $\frac{1}{\sqrt[4]{t^{5}}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Select the equivalent expression.

\newline

\left(\frac{t}{t^{7}}\right)^{\frac{5}{24}}

\newline

\sqrt[4]{t^{5}}

\newline

\sqrt[5]{t^{4}}

\newline

\frac{1}{\sqrt[5]{t^{4}}}

\newline

\frac{1}{\sqrt[4]{t^{5}}}

Simplify Base: Simplify the base of the given expression. $\newline$ We have $\left(\frac{t}{t^{7}}\right)^{\left(\frac{5}{24}\right)}$ . First, we simplify the fraction inside the parentheses by using the property of exponents that states when dividing like bases, we subtract the exponents. $\newline$ $t^1 / t^7 = t^{(1-7)} = t^{-6}.$
Apply Exponent: Apply the exponent to the simplified base. $\newline$ Now we have $(t^{-6})^{(5)/(24)}$ . We use the property of exponents that states $(a^m)^n = a^{m*n}$ to simplify the expression. $\newline$ $(t^{-6})^{(5)/(24)} = t^{(-6)*(5/24)} = t^{-5/4}$ .
Convert Negative Exponent: Convert the negative exponent to a positive exponent by taking the reciprocal. $t^{(-5/4)}$ can be rewritten as $1/(t^{(5/4)})$ . This is because a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
Rewrite as Radical: Rewrite the expression as a radical. $\newline$ $\frac{1}{t^{\frac{5}{4}}}$ can be rewritten as a radical expression. The denominator of the exponent ( $4$ ) becomes the index of the root, and the numerator ( $5$ ) becomes the exponent inside the root. $\newline$ $\frac{1}{t^{\frac{5}{4}}} = \frac{1}{\sqrt[4]{t^5}}$ .
Compare with Options: Compare the result with the given options. $\newline$ The expression we have found, $1/\sqrt[4]{t^5}$ , matches one of the given options.