The expression root(4)(5^(3))*root(5)(5^(4)) is equivalent to 25^((3)/(5)) 5^((3)/(5)) 5^((31)/(20)) 25^((31)/(20))

Understand the expression: Understand the given expression. We have the expression root(4)(5^(3))*root(5)(5^(4)), which means we are dealing with the fourth root of 5 raised to the power of 3, multiplied by the fifth root of 5 raised to the power of 4.Convert to fractional exponents: Convert the roots to fractional exponents. The fourth root of 5^3 can be written as (5^3)^(1/4), and the fifth root of 5^4 can be written as (5^4)^(1/5).Apply power of a power rule: Apply the power of a power rule. Using the power of a power rule (a^(m))^n = a^(m*n), we can simplify the expression: (5^3)^(1/4) * (5^4)^(1/5) = 5^(3/4) * 5^(4/5).Add exponents with same base: Add the exponents since the bases are the same. When multiplying with the same base, we add the exponents: 5^(3/4 + 4/5).Find common denominator: Find a common denominator to add the fractions. The common denominator for 4 and 5 is 20, so we convert the fractions: 3/4 = 15/20 and 4/5 = 16/20.Add converted fractions: Add the converted fractions. Now we add the fractions with the common denominator: 15/20 + 16/20 = 31/20.Write final expression: Write the final expression. The final expression is 5^(31/20).

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

25^((3)/(5))

5^((3)/(5))

5^((31)/(20))

25^((31)/(20))

The expression $\sqrt[4]{5^{3}} \cdot \sqrt[5]{5^{4}}$ is equivalent to $\newline$ $25^{\frac{3}{5}}$ $\newline$ $5^{\frac{3}{5}}$ $\newline$ $5^{\frac{31}{20}}$ $\newline$ $25^{\frac{31}{20}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

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

is equivalent to

\newline

25^{\frac{3}{5}}

\newline

5^{\frac{3}{5}}

\newline

5^{\frac{31}{20}}

\newline

25^{\frac{31}{20}}

Understand the expression: Understand the given expression. $\newline$ We have the expression $\sqrt[4]{5^{3}}$ $\cdot$ $\sqrt[5]{5^{4}}$ , which means we are dealing with the fourth root of $5$ raised to the power of $3$ , multiplied by the fifth root of $5$ raised to the power of $4$ .
Convert to fractional exponents: Convert the roots to fractional exponents. $\newline$ The fourth root of $5^3$ can be written as $(5^3)^{\frac{1}{4}}$ , and the fifth root of $5^4$ can be written as $(5^4)^{\frac{1}{5}}$ .
Apply power of a power rule: Apply the power of a power rule. $\newline$ Using the power of a power rule $(a^{m})^{n} = a^{m*n}$ , we can simplify the expression: $\newline$ $(5^{3})^{1/4} * (5^{4})^{1/5} = 5^{3/4} * 5^{4/5}$ .
Add exponents with same base: Add the exponents since the bases are the same. $\newline$ When multiplying with the same base, we add the exponents: $\newline$ $5^{\frac{3}{4} + \frac{4}{5}}$ .
Find common denominator: Find a common denominator to add the fractions. $\newline$ The common denominator for $4$ and $5$ is $20$ , so we convert the fractions: $\newline$ $\frac{3}{4} = \frac{15}{20}$ and $\frac{4}{5} = \frac{16}{20}$ .
Add converted fractions: Add the converted fractions. $\newline$ Now we add the fractions with the common denominator: $\newline$ $\frac{15}{20} + \frac{16}{20} = \frac{31}{20}$ .
Write final expression: Write the final expression. $\newline$ The final expression is $5^{\frac{31}{20}}$ .