The expression root(3)(5^(4))*root(5)(5^(6)) is equivalent to 5^((8)/(5)) 25^((8)/(5)) 25^((38)/(15)) 5^((38)/(15))

Understand the expression: Understand the given expression. We have the expression root(3)(5^(4))*root(5)(5^(6)), which involves cube roots and fifth roots of powers of 5.Convert to fractional exponents: Convert the roots to fractional exponents. The cube root of 5^4 can be written as 5^(4/3), and the fifth root of 5^6 can be written as 5^(6/5). So, the expression becomes 5^(4/3) * 5^(6/5).Apply multiplication rule: Apply the rule for multiplying powers with the same base. When multiplying powers with the same base, we add the exponents. So, 5^(4/3) * 5^(6/5) = 5^(4/3 + 6/5).Find common denominator: Find a common denominator to add the fractions. The common denominator for 3 and 5 is 15. So, we convert the fractions to have the denominator of 15: 4/3 = (4*5)/(3*5) = 20/15 6/5 = (6*3)/(5*3) = 18/15Add fractions: Add the fractions. Now we add the fractions with the common denominator: 20/15 + 18/15 = (20 + 18)/15 = 38/15Write final expression: Write the final expression. The expression now is 5^(38/15).Check final expression: Check if the final expression matches any of the given options. The final expression 5^(38/15) matches the last option, so the equivalent expression is 5^((38)/(15)).

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The expression
root(3)(5^(4))*root(5)(5^(6)) is equivalent to

5^((8)/(5))

25^((8)/(5))

25^((38)/(15))

5^((38)/(15))

The expression $\sqrt[3]{5^{4}} \cdot \sqrt[5]{5^{6}}$ is equivalent to $\newline$ $5^{\frac{8}{5}}$ $\newline$ $25^{\frac{8}{5}}$ $\newline$ $25^{\frac{38}{15}}$ $\newline$ $5^{\frac{38}{15}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[3]{5^{4}} \cdot \sqrt[5]{5^{6}}

is equivalent to

\newline

5^{\frac{8}{5}}

\newline

25^{\frac{8}{5}}

\newline

25^{\frac{38}{15}}

\newline

5^{\frac{38}{15}}

Understand the expression: Understand the given expression. $\newline$ We have the expression $\sqrt[3]{5^{4}}\cdot\sqrt[5]{5^{6}}$ , which involves cube roots and fifth roots of powers of $5$ .
Convert to fractional exponents: Convert the roots to fractional exponents. $\newline$ The cube root of $5^4$ can be written as $5^{\frac{4}{3}}$ , and the fifth root of $5^6$ can be written as $5^{\frac{6}{5}}$ . $\newline$ So, the expression becomes $5^{\frac{4}{3}} \times 5^{\frac{6}{5}}$ .
Apply multiplication rule: Apply the rule for multiplying powers with the same base. $\newline$ When multiplying powers with the same base, we add the exponents. $\newline$ So, $5^{\frac{4}{3}} \times 5^{\frac{6}{5}} = 5^{\frac{4}{3} + \frac{6}{5}}$ .
Find common denominator: Find a common denominator to add the fractions. $\newline$ The common denominator for $3$ and $5$ is $15$ . $\newline$ So, we convert the fractions to have the denominator of $15$ : $\newline$ $\frac{4}{3} = \left(\frac{4\times5}{3\times5}\right) = \frac{20}{15}$ $\newline$ $\frac{6}{5} = \left(\frac{6\times3}{5\times3}\right) = \frac{18}{15}$
Add fractions: Add the fractions. $\newline$ Now we add the fractions with the common denominator: $\newline$ $\frac{20}{15} + \frac{18}{15} = \frac{(20 + 18)}{15} = \frac{38}{15}$
Write final expression: Write the final expression. $\newline$ The expression now is $5^{\frac{38}{15}}$ .
Check final expression: Check if the final expression matches any of the given options. $\newline$ The final expression $5^{\frac{38}{15}}$ matches the last option, so the equivalent expression is $5^{\left(\frac{38}{15}\right)}$ .