The expression root(6)(7)*root(3)(7^(5)) is equivalent to 7^((6)/(11)) 7^((11)/(6)) 7^((18)/(5)) 7^((5)/(18))

Express Roots as Exponents: We are given the expression root(6)(7)*root(3)(7^(5)). We need to express both roots in terms of exponents. The sixth root of 7 can be written as 7^(1/6), and the cube root of 7^5 can be written as (7^5)^(1/3).Apply Exponent Rule: Now we apply the exponent rule (a^m)^n = a^(m*n) to the second term (7^5)^(1/3). This simplifies to 7^(5*(1/3)) = 7^(5/3).Multiply Expressions: Next, we multiply the two expressions together: 7^(1/6) * 7^(5/3). Since the bases are the same, we can add the exponents: 7^(1/6 + 5/3).Add Exponents: To add the exponents, we need a common denominator. The least common multiple of 6 and 3 is 6, so we convert 5/3 to an equivalent fraction with a denominator of 6: 5/3 = (5*2)/(3*2) = 10/6.Convert to Simplified Form: Now we can add the exponents: 7^(1/6 + 10/6) = 7^(11/6). This is the simplified form of the original expression.

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The expression
root(6)(7)*root(3)(7^(5)) is equivalent to

7^((6)/(11))

7^((11)/(6))

7^((18)/(5))

7^((5)/(18))

The expression $\sqrt[6]{7} \cdot \sqrt[3]{7^{5}}$ is equivalent to $\newline$ $7^{\frac{6}{11}}$ $\newline$ $7^{\frac{11}{6}}$ $\newline$ $7^{\frac{18}{5}}$ $\newline$ $7^{\frac{5}{18}}$

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

\sqrt[6]{7} \cdot \sqrt[3]{7^{5}}

is equivalent to

\newline

7^{\frac{6}{11}}

\newline

7^{\frac{11}{6}}

\newline

7^{\frac{18}{5}}

\newline

7^{\frac{5}{18}}

Express Roots as Exponents: We are given the expression $\sqrt[6]{7}\cdot\sqrt[3]{7^{5}}$ . We need to express both roots in terms of exponents. $\newline$ The sixth root of $7$ can be written as $7^{\frac{1}{6}}$ , and the cube root of $7^5$ can be written as $(7^5)^{\frac{1}{3}}$ .
Apply Exponent Rule: Now we apply the exponent rule $(a^m)^n = a^{m*n}$ to the second term $(7^5)^{1/3}$ . This simplifies to $7^{5*(1/3)} = 7^{5/3}$ .
Multiply Expressions: Next, we multiply the two expressions together: $7^{\frac{1}{6}} \times 7^{\frac{5}{3}}$ . Since the bases are the same, we can add the exponents: $7^{\frac{1}{6} + \frac{5}{3}}$ .
Add Exponents: To add the exponents, we need a common denominator. The least common multiple of $6$ and $3$ is $6$ , so we convert $\frac{5}{3}$ to an equivalent fraction with a denominator of $6$ : $\frac{5}{3} = \frac{(5\times2)}{(3\times2)} = \frac{10}{6}$ .
Convert to Simplified Form: Now we can add the exponents: $7^{\frac{1}{6} + \frac{10}{6}} = 7^{\frac{11}{6}}$ . $\newline$ This is the simplified form of the original expression.