The quotient of (root(5)(m^(4)))/(root(4)(m^(3))) and (sqrtm)/(root(5)(m^(6))) is equal to m^(y). What is the value of y ?

Question

The quotient of (root(5)(m^(4)))/(root(4)(m^(3))) and (sqrtm)/(root(5)(m^(6))) is equal to  m^(y). What is the value of y ?

Bytelearn · Accepted Answer

Convert roots to exponents: We need to find the quotient of two expressions: (root(5)(m^(4)))/(root(4)(m^(3))) and (sqrt(m))/(root(5)(m^(6))). 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). 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^(4/5 - 3/4). To subtract these fractions, we need a common denominator, which is 20. So, we rewrite the exponents as m^(16/20 - 15/20) = m^(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). 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^(1/20) / m^(-7/10). To divide these, we subtract the exponents: m^(1/20 - (-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 15/20 by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, m^(15/20) simplifies to m^(3/4).Final result: We have found that the quotient of the two expressions simplifies to m^(3/4). Therefore, the value of y in the expression m^(y) is 3/4.

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 $y$ ?

Full solution

More problems from Simplify radical expressions with variables

Related topics

The quotient of m45m34 \frac{\sqrt[5]{m^{4}}}{\sqrt[4]{m^{3}}} 4m3​5m4​​ and mm65 \frac{\sqrt{m}}{\sqrt[5]{m^{6}}} 5m6​m​​ is equal to my m^{y} my. What is the value of y y y ?

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 $y$ ?