Question

What is the number for the expression once it is simplified? This should not have any exponents. ((64x-⁹y-¹⁵) ⅓)/ ((8²x-¹²y¹⁰)½)

Answer 1

The expression to simplify is:

$\frac{(64x^{-9}y^{-15})^{\frac{1}{3}}}{(8^2x^{-12}y^{10})^{\frac{1}{2}}}$

We can use the property shown below to simplify it:

$(a^m)^n=a^{mn}$

Thus, we can write:

$\begin{gathered} \frac{(64x^{-9}y^{-15})^{\frac{1}{3}}}{(8^2x^{-12}y^{10})^{\frac{1}{2}}} \\ =\frac{(64)^{\frac{1}{3}}(x^{-9})^{\frac{1}{3}}(y^{-15})^{\frac{1}{3}}}{(64)^{\frac{1}{2}}(x^{-12})^{\frac{1}{2}}(y^{10})^{\frac{1}{2}}} \end{gathered}$

64 to 1/3rd power is 4

64 to 1/2 power is 8

Thus, further simplifying, we have:

$\begin{gathered} \frac{(64)^{\frac{1}{3}}(x^{-9})^{\frac{1}{3}}(y^{-15})^{\frac{1}{3}}}{(64)^{\frac{1}{2}}(x^{-12})^{\frac{1}{2}}(y^{10})^{\frac{1}{2}}} \\ =\frac{4x^{-3}y^{-5}}{8x^{-6}y^5} \end{gathered}$

We can use the rule

$\frac{x^a}{x^b}=x^{a-b}$

to bring up all the exponents and get the simplified form:

$\begin{gathered} =\frac{x^{-3}y^{-5}x^6y^{-5}}{2} \\ =\frac{x^3y^{-10}}{2} \end{gathered}$

This is the final answer.

The exponent of x is 3 and the exponent of y is -10.