The expression root(5)(7^(4))*root(6)(7^(5)) is equivalent to 7^((3)/(2)) 7^((49)/(30)) 7^((30)/(49)) 7^((2)/(3))

Understand the problem: Understand the problem. We need to simplify the expression involving roots of powers of 7.Convert to fractional exponents: Convert roots to fractional exponents. root(5)(7^(4)) can be written as 7^(4/5) and root(6)(7^(5)) can be written as 7^(5/6).Multiply expressions: Multiply the expressions with fractional exponents. 7^(4/5) * 7^(5/6) = 7^(4/5 + 5/6)Find common denominator: Find a common denominator for the exponents. The common denominator for 5 and 6 is 30. So we convert the fractions to have the same denominator: 4/5 = (4*6)/(5*6) = 24/30 5/6 = (5*5)/(6*5) = 25/30Add exponents: Add the exponents. 7^(24/30 + 25/30) = 7^(49/30)Write final answer: Write the final answer. The expression root(5)(7^(4))*root(6)(7^(5)) is equivalent to 7^(49/30).

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The expression
root(5)(7^(4))*root(6)(7^(5)) is equivalent to

7^((3)/(2))

7^((49)/(30))

7^((30)/(49))

7^((2)/(3))

The expression $\sqrt[5]{7^{4}} \cdot \sqrt[6]{7^{5}}$ is equivalent to $\newline$ $7^{\frac{3}{2}}$ $\newline$ $7^{\frac{49}{30}}$ $\newline$ $7^{\frac{30}{49}}$ $\newline$ $7^{\frac{2}{3}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[5]{7^{4}} \cdot \sqrt[6]{7^{5}}

is equivalent to

\newline

7^{\frac{3}{2}}

\newline

7^{\frac{49}{30}}

\newline

7^{\frac{30}{49}}

\newline

7^{\frac{2}{3}}

Understand the problem: Understand the problem. We need to simplify the expression involving roots of powers of $7$ .
Convert to fractional exponents: Convert roots to fractional exponents. $\sqrt[5]{7^{4}}$ can be written as $7^{\frac{4}{5}}$ and $\sqrt[6]{7^{5}}$ can be written as $7^{\frac{5}{6}}$ .
Multiply expressions: Multiply the expressions with fractional exponents. $\newline$ $7^{\frac{4}{5}} \times 7^{\frac{5}{6}} = 7^{\frac{4}{5} + \frac{5}{6}}$
Find common denominator: Find a common denominator for the exponents. $\newline$ The common denominator for $5$ and $6$ is $30$ . So we convert the fractions to have the same denominator: $\newline$ $\frac{4}{5} = \left(\frac{4\times6}{5\times6}\right) = \frac{24}{30}$ $\newline$ $\frac{5}{6} = \left(\frac{5\times5}{6\times5}\right) = \frac{25}{30}$
Add exponents: Add the exponents. $\newline$ $7^{\frac{24}{30} + \frac{25}{30}} = 7^{\frac{49}{30}}$
Write final answer: Write the final answer. $\newline$ The expression $\sqrt[5]{7^{4}}$ $\sqrt[6]{7^{5}}$ is equivalent to $7^{\frac{49}{30}}$ .