The expression root(4)(2^(3))*root(3)(2^(4)) is equivalent to 4 2^((25)/(12)) 2 4^((25)/(12))

Understand Expression: Understand the given expression. We have the expression root(4)(2^(3))*root(3)(2^(4)), which involves two different roots of powers of 2.Convert to Exponents: Convert roots to fractional exponents. The fourth root of 2^3 can be written as (2^3)^(1/4), and the cube root of 2^4 can be written as (2^4)^(1/3).Apply Power Rule: Apply the power of a power rule. Using the power of a power rule, we simplify the exponents: (2^3)^(1/4) = 2^(3*(1/4)) = 2^(3/4) and (2^4)^(1/3) = 2^(4*(1/3)) = 2^(4/3).Multiply with Same Base: Multiply the expressions with the same base. Since we are multiplying two expressions with the same base (2), we can add the exponents: 2^(3/4) * 2^(4/3).Add Exponents: Add the exponents. To add the exponents, we find a common denominator, which is 12: (3/4) * (3/3) = 9/12 and (4/3) * (4/4) = 16/12. Now we add the fractions: 9/12 + 16/12 = 25/12.Write Final Expression: Write the final expression. The final expression after adding the exponents is 2^(25/12).Check Given Options: Check if the expression matches any of the given options. The expression 2^(25/12) matches the second option, which is 2^((25)/(12)).

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

2^((25)/(12))
2

4^((25)/(12))

The expression $\sqrt[4]{2^{3}} \cdot \sqrt[3]{2^{4}}$ is equivalent to $\newline$ $4$ $\newline$ $2^{\frac{25}{12}}$ $\newline$ $2$ $\newline$ $4^{\frac{25}{12}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

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

is equivalent to

\newline

4

\newline

2^{\frac{25}{12}}

\newline

2

\newline

4^{\frac{25}{12}}

Understand Expression: Understand the given expression. $\newline$ We have the expression $\sqrt[4]{2^{3}}$ $\cdot$ $\sqrt[3]{2^{4}}$ , which involves two different roots of powers of $2$ .
Convert to Exponents: Convert roots to fractional exponents. $\newline$ The fourth root of $2^3$ can be written as $(2^3)^{\frac{1}{4}}$ , and the cube root of $2^4$ can be written as $(2^4)^{\frac{1}{3}}$ .
Apply Power Rule: Apply the power of a power rule. $\newline$ Using the power of a power rule, we simplify the exponents: $(2^3)^{1/4} = 2^{3*(1/4)} = 2^{3/4}$ and $(2^4)^{1/3} = 2^{4*(1/3)} = 2^{4/3}$ .
Multiply with Same Base: Multiply the expressions with the same base. $\newline$ Since we are multiplying two expressions with the same base $2$ , we can add the exponents: $2^{3/4} \times 2^{4/3}$ .
Add Exponents: Add the exponents. $\newline$ To add the exponents, we find a common denominator, which is $12$ : $(3/4) \times (3/3) = 9/12$ and $(4/3) \times (4/4) = 16/12$ . Now we add the fractions: $9/12 + 16/12 = 25/12$ .
Write Final Expression: Write the final expression. $\newline$ The final expression after adding the exponents is $2^{\frac{25}{12}}$ .
Check Given Options: Check if the expression matches any of the given options. $\newline$ The expression $2^{\frac{25}{12}}$ matches the second option, which is $2^{\left(\frac{25}{12}\right)}$ .