Question

find the product. assume variables represent positive real numbers.
$(m^{1/8} + m^{-1/8})(m^{-1/4} - m^{1/4})$
$\bigcirc -m^{1/4} + m^{-1/4}$
$\bigcirc m - m^{1/4} + m^{-1/4} - 1$
$\bigcirc 1 - m^{1/4} + m^{-1/4} - m$
$\bigcirc m^{1/4} - m^{-1/4}$

Explanation:

Step1: Apply difference of squares

Recall $(a+b)(a-b)=a^2-b^2$. Let $a=m^{-1/8}$, $b=m^{1/8}$.
$$(m^{1/8}+m^{-1/8})(m^{-1/8}-m^{1/8})=(m^{-1/8})^2-(m^{1/8})^2$$

Step2: Simplify exponents

Use $(x^n)^m=x^{n\cdot m}$.
$$(m^{-1/8})^2 = m^{-\frac{1}{8}\cdot2}=m^{-1/4}$$
$$(m^{1/8})^2 = m^{\frac{1}{8}\cdot2}=m^{1/4}$$

Step3: Rewrite the expression

Substitute back to get the final form.
$$m^{-1/4}-m^{1/4}=-m^{1/4}+m^{-1/4}$$

Answer:

$-m^{1/4} + m^{-1/4}$