The expression root(5)(7^(4))*root(3)(7^(2)) is equivalent to 7^((15)/(22)) 7^((15)/(8)) 7^((8)/(15)) 7^((22)/(15))

Understand Expression: Understand the given expression. We have the expression root(5)(7^(4))*root(3)(7^(2)), which involves two different roots of the same base number 7 raised to different powers.Convert to Fractional Exponents: Convert the roots to fractional exponents. The 5th root of 7^4 can be written as 7^(4/5), and the cube root of 7^2 can be written as 7^(2/3). So, the expression becomes 7^(4/5) * 7^(2/3).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, 7^(4/5) * 7^(2/3) becomes 7^(4/5 + 2/3).Find Common Denominator: Find a common denominator to add the fractions. The common denominator for 5 and 3 is 15. Convert the fractions to have the same denominator. 4/5 = (4*3)/(5*3) = 12/15 2/3 = (2*5)/(3*5) = 10/15 Now, add the fractions: 12/15 + 10/15 = 22/15.Write Final Expression: Write the final expression. The expression now is 7^(22/15).

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The expression
root(5)(7^(4))*root(3)(7^(2)) is equivalent to

7^((15)/(22))

7^((15)/(8))

7^((8)/(15))

7^((22)/(15))

The expression $\sqrt[5]{7^{4}} \cdot \sqrt[3]{7^{2}}$ is equivalent to $\newline$ $7^{\frac{15}{22}}$ $\newline$ $7^{\frac{15}{8}}$ $\newline$ $7^{\frac{8}{15}}$ $\newline$ $7^{\frac{22}{15}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[5]{7^{4}} \cdot \sqrt[3]{7^{2}}

is equivalent to

\newline

7^{\frac{15}{22}}

\newline

7^{\frac{15}{8}}

\newline

7^{\frac{8}{15}}

\newline

7^{\frac{22}{15}}

Understand Expression: Understand the given expression. $\newline$ We have the expression $\sqrt[5]{7^{4}}$ $\cdot$ $\sqrt[3]{7^{2}}$ , which involves two different roots of the same base number $7$ raised to different powers.
Convert to Fractional Exponents: Convert the roots to fractional exponents. $\newline$ The $5^{\text{th}}$ root of $7^4$ can be written as $7^{\frac{4}{5}}$ , and the cube root of $7^2$ can be written as $7^{\frac{2}{3}}$ . $\newline$ So, the expression becomes $7^{\frac{4}{5}} \times 7^{\frac{2}{3}}$ .
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, $7^{\frac{4}{5}} \times 7^{\frac{2}{3}}$ becomes $7^{\frac{4}{5} + \frac{2}{3}}$ .
Find Common Denominator: Find a common denominator to add the fractions. $\newline$ The common denominator for $5$ and $3$ is $15$ . Convert the fractions to have the same denominator. $\newline$ $\frac{4}{5} = \left(\frac{4\times3}{5\times3}\right) = \frac{12}{15}$ $\newline$ $\frac{2}{3} = \left(\frac{2\times5}{3\times5}\right) = \frac{10}{15}$ $\newline$ Now, add the fractions: $\frac{12}{15} + \frac{10}{15} = \frac{22}{15}$ .
Write Final Expression: Write the final expression. $\newline$ The expression now is $7^{\frac{22}{15}}$ .