The expression root(5)(2^(4))*root(5)(2^(6)) is equivalent to 2^(2) 2^((24)/(25)) 4^(2) 4^((24)/(25))

Understand the Problem: Understand the problem. We need to simplify the expression involving fifth roots of powers of 2.Apply Exponent Property: Apply the property of exponents for roots. The fifth root of a number is the same as raising that number to the power of 1/5. So, root(5)(2^(4)) is the same as (2^(4))^(1/5) and root(5)(2^(6)) is the same as (2^(6))^(1/5).Use Power Rule: Use the power of a power rule. When you raise a power to a power, you multiply the exponents. (2^(4))^(1/5) * (2^(6))^(1/5) = 2^(4/5) * 2^(6/5)Combine Terms: Combine the terms with the same base. When multiplying terms with the same base, add the exponents. 2^(4/5) * 2^(6/5) = 2^((4/5) + (6/5)) = 2^((4+6)/5) = 2^(10/5)Simplify Exponent: Simplify the exponent. 2^(10/5) simplifies to 2^2 because 10 divided by 5 is 2. 2^(10/5) = 2^2Check Answer Choices: Check the answer choices. The correct answer is 2^2, which matches one of the given options.

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The expression
root(5)(2^(4))*root(5)(2^(6)) is equivalent to

2^(2)

2^((24)/(25))

4^(2)

4^((24)/(25))

The expression $\sqrt[5]{2^{4}} \cdot \sqrt[5]{2^{6}}$ is equivalent to $\newline$ $2^{2}$ $\newline$ $2^{\frac{24}{25}}$ $\newline$ $4^{2}$ $\newline$ $4^{\frac{24}{25}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

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Q. The expression

\sqrt[5]{2^{4}} \cdot \sqrt[5]{2^{6}}

is equivalent to

\newline

2^{2}

\newline

2^{\frac{24}{25}}

\newline

4^{2}

\newline

4^{\frac{24}{25}}

Understand the Problem: Understand the problem. We need to simplify the expression involving fifth roots of powers of $2$ .
Apply Exponent Property: Apply the property of exponents for roots. The fifth root of a number is the same as raising that number to the power of $\frac{1}{5}$ . So, $\sqrt[5]{2^{4}}$ is the same as $(2^{4})^{\frac{1}{5}}$ and $\sqrt[5]{2^{6}}$ is the same as $(2^{6})^{\frac{1}{5}}$ .
Use Power Rule: Use the power of a power rule. $\newline$ When you raise a power to a power, you multiply the exponents. $\newline$ $(2^{4})^{1/5} \times (2^{6})^{1/5} = 2^{4/5} \times 2^{6/5}$
Combine Terms: Combine the terms with the same base. $\newline$ When multiplying terms with the same base, add the exponents. $\newline$ $2^{\frac{4}{5}} \times 2^{\frac{6}{5}} = 2^{\left(\frac{4}{5} + \frac{6}{5}\right)} = 2^{\left(\frac{4+6}{5}\right)} = 2^{\frac{10}{5}}$
Simplify Exponent: Simplify the exponent. $2^{(10/5)}$ simplifies to $2^2$ because $10$ divided by $5$ is $2$ . $2^{(10/5)} = 2^2$
Check Answer Choices: Check the answer choices. $\newline$ The correct answer is $2^2$ , which matches one of the given options.