The expression root(3)(2^(4))*root(3)(2^(4)) is equivalent to 2^((16)/(9)) 4^((8)/(3)) 4^((16)/(9)) 2^((8)/(3))

Understand the problem: Understand the problem. We need to simplify the expression root(3)(2^(4))*root(3)(2^(4)).Rewrite using radical notation: Rewrite the expression using radical notation. root(3)(2^(4)) can be written as (2^(4))^(1/3). So the expression becomes (2^(4))^(1/3) * (2^(4))^(1/3).Apply power of a power rule: Apply the power of a power rule. When you raise a power to a power, you multiply the exponents. So (2^(4))^(1/3) * (2^(4))^(1/3) becomes 2^(4*(1/3)) * 2^(4*(1/3)).Simplify the exponents: Simplify the exponents. 4*(1/3) is 4/3. So we have 2^(4/3) * 2^(4/3).Apply product of powers rule: Apply the product of powers rule. When you multiply powers with the same base, you add the exponents. So 2^(4/3) * 2^(4/3) becomes 2^((4/3) + (4/3)).Add the exponents: Add the exponents. (4/3) + (4/3) is 8/3. So the expression simplifies to 2^(8/3).

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The expression
root(3)(2^(4))*root(3)(2^(4)) is equivalent to

2^((16)/(9))

4^((8)/(3))

4^((16)/(9))

2^((8)/(3))

The expression $\sqrt[3]{2^{4}} \cdot \sqrt[3]{2^{4}}$ is equivalent to $\newline$ $2^{\frac{16}{9}}$ $\newline$ $4^{\frac{8}{3}}$ $\newline$ $4^{\frac{16}{9}}$ $\newline$ $2^{\frac{8}{3}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[3]{2^{4}} \cdot \sqrt[3]{2^{4}}

is equivalent to

\newline

2^{\frac{16}{9}}

\newline

4^{\frac{8}{3}}

\newline

4^{\frac{16}{9}}

\newline

2^{\frac{8}{3}}

Understand the problem: Understand the problem. $\newline$ We need to simplify the expression $\sqrt[3]{2^{4}}$ $\sqrt[3]{2^{4}}$ .
Rewrite using radical notation: Rewrite the expression using radical notation. $\newline$ $\sqrt[3]{2^{4}}$ can be written as $(2^{4})^{\frac{1}{3}}$ . $\newline$ So the expression becomes $(2^{4})^{\frac{1}{3}} \times (2^{4})^{\frac{1}{3}}$ .
Apply power of a power rule: Apply the power of a power rule. $\newline$ When you raise a power to a power, you multiply the exponents. $\newline$ So $(2^{4})^{\frac{1}{3}} \times (2^{4})^{\frac{1}{3}}$ becomes $2^{4\times(\frac{1}{3})} \times 2^{4\times(\frac{1}{3})}$ .
Simplify the exponents: Simplify the exponents. $\newline$ $4\times\left(\frac{1}{3}\right)$ is $\frac{4}{3}$ . $\newline$ So we have $2^{\frac{4}{3}} \times 2^{\frac{4}{3}}$ .
Apply product of powers rule: Apply the product of powers rule. $\newline$ When you multiply powers with the same base, you add the exponents. $\newline$ So $2^{\frac{4}{3}} \times 2^{\frac{4}{3}}$ becomes $2^{(\frac{4}{3} + \frac{4}{3})}$ .
Add the exponents: Add the exponents. $\newline$ $(\frac{4}{3}) + (\frac{4}{3})$ is $\frac{8}{3}$ . $\newline$ So the expression simplifies to $2^{\frac{8}{3}}$ .