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

Express Radicals as Exponents: First, let's express the radicals as exponents and write the given expression. root(5)(m^(4)) can be written as m^(4/5). root(4)(m^(3)) can be written as m^(3/4). sqrt(m) can be written as m^(1/2). root(5)(m^(6)) can be written as m^(6/5).  So the expression becomes: (m^(4/5) / m^(3/4)) / (m^(1/2) / m^(6/5))Simplify Using Quotient Rule: Next, we simplify the expression by using the quotient rule of exponents, which states that a^(m)/a^(n) = a^(m-n).  For the numerator: m^(4/5) / m^(3/4) = m^(4/5 - 3/4) For the denominator: m^(1/2) / m^(6/5) = m^(1/2 - 6/5)  Now we need to find a common denominator to subtract the fractions in the exponents.Find Common Denominator: For the numerator, the common denominator for 5 and 4 is 20. So we convert the exponents: m^(4/5 - 3/4) = m^(16/20 - 15/20) = m^(1/20)  For the denominator, the common denominator for 2 and 5 is 10. So we convert the exponents: m^(1/2 - 6/5) = m^(5/10 - 12/10) = m^(-7/10)  Now we have: (m^(1/20)) / (m^(-7/10))Further Simplify Expression: We can simplify the expression further by using the property that a^(m)/a^(n) = a^(m-n) again.  So we have: m^(1/20 - (-7/10)) = m^(1/20 + 7/10)  Now we need to find a common denominator to add the fractions in the exponents.Add Exponents: The common denominator for 20 and 10 is 20. So we convert the exponents: m^(1/20 + 7/10) = m^(1/20 + 14/20) = m^(15/20)  Now we simplify the fraction: m^(15/20) = m^(3/4)  So the expression simplifies to m^(3/4).Equate Exponents: Since the original expression is equal to m^(y), we can now equate the exponents: y = 3/4  This gives us the value of 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 $y$ ?

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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 $y$ ?