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

Express in Exponents: We are given the expression sqrt(7^(3))*root(6)(7^(5)). The first step is to express the square root and the sixth root in terms of exponents. The square root of a number is the same as raising that number to the power of 1/2, and the sixth root is the same as raising that number to the power of 1/6. So, we can rewrite the expression as: 7^(3/2) * 7^(5/6)Apply Exponent Rule: Next, we apply the rule of exponents that states when we multiply two exponents with the same base, we add the exponents. So, we add the exponents 3/2 and 5/6. To add these fractions, we need a common denominator, which is 6 in this case. So, we convert 3/2 to 9/6 and then add it to 5/6. 9/6 + 5/6 = 14/6Simplify Fraction: Now, we simplify the fraction 14/6 by dividing both the numerator and the denominator by their greatest common divisor, which is 2. 14 ÷ 2 = 7 6 ÷ 2 = 3 So, 14/6 simplifies to 7/3.Write Final Expression: Finally, we write the simplified exponent back with the base 7. The expression becomes 7^(7/3).

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

7^((4)/(5))

7^((5)/(4))

7^((7)/(3))

7^((3)/(7))

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

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. The expression

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

is equivalent to

\newline

7^{\frac{4}{5}}

\newline

7^{\frac{5}{4}}

\newline

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

\newline

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

Express in Exponents: We are given the expression $\sqrt{7^{3}}\cdot\sqrt[6]{7^{5}}$ . The first step is to express the square root and the sixth root in terms of exponents. $\newline$ The square root of a number is the same as raising that number to the power of $\frac{1}{2}$ , and the sixth root is the same as raising that number to the power of $\frac{1}{6}$ . $\newline$ So, we can rewrite the expression as: $\newline$ $7^{\frac{3}{2}} \cdot 7^{\frac{5}{6}}$
Apply Exponent Rule: Next, we apply the rule of exponents that states when we multiply two exponents with the same base, we add the exponents. $\newline$ So, we add the exponents $\frac{3}{2}$ and $\frac{5}{6}$ . $\newline$ To add these fractions, we need a common denominator, which is $6$ in this case. $\newline$ So, we convert $\frac{3}{2}$ to $\frac{9}{6}$ and then add it to $\frac{5}{6}$ . $\newline$ $\frac{9}{6} + \frac{5}{6} = \frac{14}{6}$
Simplify Fraction: Now, we simplify the fraction $\frac{14}{6}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $2$ . $\newline$ $14 \div 2 = 7$ $\newline$ $6 \div 2 = 3$ $\newline$ So, $\frac{14}{6}$ simplifies to $\frac{7}{3}$ .
Write Final Expression: Finally, we write the simplified exponent back with the base $7$ . The expression becomes $7^{7/3}$ .