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$Rewrite the expression in the form k*y^(n). Write the exponent as an integer, fraction, or an exact decimal (not a mixed number). (4root(4)(y^(5)))^((1)/(2))=◻$

Rewrite the expression in the form $k \cdot y^{n}$ . $\newline$ Write the exponent as an integer, fraction, or an exact decimal (not a mixed number). $\newline$ $\left(4 \sqrt[4]{y^{5}}\right)^{\frac{1}{2}}=\square$

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Powers with decimal and fractional bases

Full solution

Q. Rewrite the expression in the form

k \cdot y^{n}

.

\newline

Write the exponent as an integer, fraction, or an exact decimal (not a mixed number).

\newline

\left(4 \sqrt[4]{y^{5}}\right)^{\frac{1}{2}}=\square

Identify given expression and operations: Identify the given expression and the operations involved. $\newline$ The given expression is $(4\sqrt[4]{y^{5}})^{\frac{1}{2}}$ , which involves a fourth root and an exponentiation by $\frac{1}{2}$ .
Rewrite fourth root as exponent: Rewrite the fourth root as an exponent. $\newline$ The fourth root of $y^{5}$ can be written as $y^{\frac{5}{4}}$ because the fourth root is equivalent to raising to the power of $\frac{1}{4}$ .
Apply exponent to previous result: Apply the exponent of $\frac{1}{2}$ to the result from the previous step. $\newline$ When we raise $y^{\frac{5}{4}}$ to the power of $\frac{1}{2}$ , we multiply the exponents: $\left(\frac{5}{4}\right) \times \left(\frac{1}{2}\right) = \frac{5}{8}$ .
Combine steps to write expression: Combine the steps to write the expression in the form $k*y^{n}$ . Since there is no coefficient other than $1$ in front of the $y$ term, the expression in the required form is $y^{\frac{5}{8}}$ .

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