Write the Fraction in Simplest Form ((5k^3b^2)^-2)/(5k^7)

$\frac{{(5 k^{3} b^{2})}^{- 2}}{5 k^{7}}$

Step 1

Move ${(5 k^{3} b^{2})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{5 k^{7} {(5 k^{3} b^{2})}^{2}}$

Step 2

Simplify the denominator.

Apply the product rule to $5 k^{3} b^{2}$ .

$\frac{1}{5 k^{7} {(5 k^{3})}^{2} {(b^{2})}^{2}}$

Apply the product rule to $5 k^{3}$ .

$\frac{1}{5 k^{7} (5^{2} {(k^{3})}^{2}) {(b^{2})}^{2}}$

Raise $5$ to the power of $2$ .

$\frac{1}{5 k^{7} (25 {(k^{3})}^{2}) {(b^{2})}^{2}}$

Multiply the exponents in ${(k^{3})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{5 k^{7} (25 k^{3 \cdot 2}) {(b^{2})}^{2}}$

Multiply $3$ by $2$ .

$\frac{1}{5 k^{7} (25 k^{6}) {(b^{2})}^{2}}$

Multiply the exponents in ${(b^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{5 k^{7} (25 k^{6}) b^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{1}{5 k^{7} (25 k^{6}) b^{4}}$

$\frac{1}{5 k^{7} \cdot 25 k^{6} b^{4}}$

Combine exponents.

Multiply $25$ by $5$ .

$\frac{1}{125 k^{7} k^{6} b^{4}}$

Multiply $k^{7}$ by $k^{6}$ by adding the exponents.

Move $k^{6}$ .

$\frac{1}{125 (k^{6} k^{7}) b^{4}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{125 k^{6 + 7} b^{4}}$

Add $6$ and $7$ .

$\frac{1}{125 k^{13} b^{4}}$