Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{2h^{7}v^{-3}}{(5h^{-4}v^{5})^{-5}}\\)

Explanation:

Step1: Simplify denominator's negative exponent

Use rule $(x^a)^b=x^{ab}$ and $x^{-n}=\frac{1}{x^n}$:
$$(5h^{-4}v^5)^{-5} = 5^{-5}h^{(-4)(-5)}v^{(5)(-5)} = \frac{1}{5^5}h^{20}v^{-25} = \frac{h^{20}}{5^5v^{25}}$$

Step2: Rewrite division as multiplication by reciprocal

$$\frac{2h^7v^{-3}}{\frac{h^{20}}{5^5v^{25}}} = 2h^7v^{-3} \times \frac{5^5v^{25}}{h^{20}}$$

Step3: Compute constants and combine like terms

Calculate $5^5=3125$, use $x^a \times x^b=x^{a+b}$ and $x^a/x^b=x^{a-b}$:
$$2 \times 3125 \times h^{7-20} \times v^{-3+25} = 6250h^{-13}v^{22}$$

Step4: Convert negative exponent to positive

Use $x^{-n}=\frac{1}{x^n}$:
$$6250 \times \frac{v^{22}}{h^{13}} = \frac{6250v^{22}}{h^{13}}$$

Answer:

$\frac{6250v^{22}}{h^{13}}$