Question

ack
feb 10, 2026 11:58pm
59pm
ent.
express the following fraction in simplest form, using only positive exponents.
$\frac{10k^{-10}b^{-9}}{5(k^{-5}b^{2})^{-3}}$

Explanation:

Step1: Simplify denominator's exponent

Use power rule: $(x^a y^b)^c = x^{ac}y^{bc}$

$$\begin{align} 5(k^{-5}b^2)^{-3}&=5k^{(-5)(-3)}b^{(2)(-3)}\\ &=5k^{15}b^{-6} \end{align}$$

Step2: Rewrite the original fraction

Substitute simplified denominator
$\frac{10k^{-10}b^{-9}}{5k^{15}b^{-6}}$

Step3: Simplify coefficients

Divide 10 by 5
$\frac{10}{5}=2$

Step4: Simplify $k$ terms

Use quotient rule: $\frac{x^a}{x^b}=x^{a-b}$
$k^{-10-15}=k^{-25}$

Step5: Simplify $b$ terms

Use quotient rule: $\frac{x^a}{x^b}=x^{a-b}$
$b^{-9-(-6)}=b^{-3}$

Step6: Combine all simplified terms

Multiply coefficients and variables
$2k^{-25}b^{-3}$

Step7: Rewrite with positive exponents

Use $x^{-a}=\frac{1}{x^a}$
$2\cdot\frac{1}{k^{25}}\cdot\frac{1}{b^{3}}=\frac{2}{k^{25}b^{3}}$

Answer:

$\frac{2}{k^{25}b^{3}}$