Google Classroom $\newline$ The quotient of $\frac{\sqrt[5]{m^{4}}}{\sqrt[4]{m^{3}}}$ and $\frac{\sqrt{m}}{\sqrt[5]{m^{6}}}$ is equal to $m^{y}$ . What is the value of $y$ ? $\newline$ Show calculator $\newline$ How do I enter a student-produced response on the SAT? [Show me!] $\newline$ Stuck? Use a hint. $\newline$ Report a problem

View step-by-step help

Home

Math Problems

Precalculus

Simplify radical expressions with variables

Full solution

Q. Google Classroom

\newline

The quotient of

\frac{\sqrt[5]{m^{4}}}{\sqrt[4]{m^{3}}}

and

\frac{\sqrt{m}}{\sqrt[5]{m^{6}}}

is equal to

m^{y}

. What is the value of

y

\newline

Show calculator

\newline

How do I enter a student-produced response on the SAT? [Show me!]

\newline

Stuck? Use a hint.

\newline

Report a problem

Convert roots to exponents: We need to find the quotient of two expressions: $\left(\sqrt[5]{m^{4}}\right)/\left(\sqrt[4]{m^{3}}\right)$ and $\left(\sqrt{m}\right)/\left(\sqrt[5]{m^{6}}\right)$ . First, let's simplify each expression separately by converting the roots to fractional exponents.
Simplify first expression: The fifth root of $m^4$ can be written as $m^{(4/5)}$ . Similarly, the fourth root of $m^3$ can be written as $m^{(3/4)}$ . $\newline$ So, the first expression becomes $m^{(4/5)} / m^{(3/4)}$ .
Subtract exponents: To divide two expressions with the same base, we subtract the exponents: $m^{\frac{4}{5} - \frac{3}{4}}$ . To subtract these fractions, we need a common denominator, which is $20$ . So, we rewrite the exponents as $m^{\frac{16}{20} - \frac{15}{20}} = m^{\frac{1}{20}}$ .
Simplify second expression: Now, let's simplify the second expression. The square root of $m$ is $m^{1/2}$ , and the fifth root of $m^6$ is $m^{6/5}$ . $\newline$ So, the second expression becomes $m^{1/2} / m^{6/5}$ .
Subtract exponents: Again, to divide two expressions with the same base, we subtract the exponents: $m^{(1/2 - 6/5)}$ . We need a common denominator, which is $10$ . So, we rewrite the exponents as $m^{(5/10 - 12/10)} = m^{(-7/10)}$ .
Find quotient: Now we need to find the quotient of the two simplified expressions: $m^{\frac{1}{20}} / m^{-\frac{7}{10}}$ . To divide these, we subtract the exponents: $m^{\frac{1}{20} - (-\frac{7}{10})}$ .
Subtract exponents: We need a common denominator to subtract these fractions, which is $20$ . So, we rewrite the exponents as $m^{(1/20) - (-14/20)} = m^{1/20 + 14/20} = m^{15/20}$ .
Simplify exponent: We can simplify the exponent $\frac{15}{20}$ by dividing both the numerator and the denominator by their greatest common divisor, which is $5$ . So, $m^{\frac{15}{20}}$ simplifies to $m^{\frac{3}{4}}$ .
Final result: We have found that the quotient of the two expressions simplifies to $m^{\frac{3}{4}}$ . Therefore, the value of $y$ in the expression $m^{y}$ is $\frac{3}{4}$ .