Simplify (5^-6x^-1y^2)/(5^-3x^-4y^4)

$\frac{5^{- 6} x^{- 1} y^{2}}{5^{- 3} x^{- 4} y^{4}}$

Step 1

Move $5^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x^{- 1} y^{2}}{5^{- 3} x^{- 4} y^{4} \cdot 5^{6}}$

Step 2

Multiply $5^{- 3}$ by $5^{6}$ by adding the exponents.

Move $5^{6}$ .

$\frac{x^{- 1} y^{2}}{5^{6} \cdot 5^{- 3} x^{- 4} y^{4}}$

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

$\frac{x^{- 1} y^{2}}{5^{6 - 3} x^{- 4} y^{4}}$

Subtract $3$ from $6$ .

$\frac{x^{- 1} y^{2}}{5^{3} x^{- 4} y^{4}}$

Step 3

Move $x^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{x^{- 1} y^{2} x^{4}}{5^{3} y^{4}}$

Step 4

Multiply $x^{- 1}$ by $x^{4}$ by adding the exponents.

Move $x^{4}$ .

$\frac{x^{4} x^{- 1} y^{2}}{5^{3} y^{4}}$

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

$\frac{x^{4 - 1} y^{2}}{5^{3} y^{4}}$

Subtract $1$ from $4$ .

$\frac{x^{3} y^{2}}{5^{3} y^{4}}$

Step 5

Cancel the common factor of $y^{2}$ and $y^{4}$ .

Factor $y^{2}$ out of $x^{3} y^{2}$ .

$\frac{y^{2} x^{3}}{5^{3} y^{4}}$

Cancel the common factors.

Factor $y^{2}$ out of $5^{3} y^{4}$ .

$\frac{y^{2} x^{3}}{y^{2} (5^{3} y^{2})}$

Cancel the common factor.

$\frac{y^{2} x^{3}}{y^{2} (5^{3} y^{2})}$

Rewrite the expression.

$\frac{x^{3}}{5^{3} y^{2}}$

Step 6

Raise $5$ to the power of $3$ .

$\frac{x^{3}}{125 y^{2}}$