Simplify (15a^2y^-3)^4(5ay^-2)^-5

${(15 a^{2} y^{- 3})}^{4} {(5 a y^{- 2})}^{- 5}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(15 a^{2} \frac{1}{y^{3}})}^{4} {(5 a y^{- 2})}^{- 5}$

Step 2

Multiply $15 a^{2} \frac{1}{y^{3}}$ .

Combine $15$ and $\frac{1}{y^{3}}$ .

${(a^{2} \frac{15}{y^{3}})}^{4} {(5 a y^{- 2})}^{- 5}$

Combine $a^{2}$ and $\frac{15}{y^{3}}$ .

${(\frac{a^{2} \cdot 15}{y^{3}})}^{4} {(5 a y^{- 2})}^{- 5}$

Step 3

Move $15$ to the left of $a^{2}$ .

${(\frac{15 \cdot a^{2}}{y^{3}})}^{4} {(5 a y^{- 2})}^{- 5}$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{15 a^{2}}{y^{3}}$ .

$\frac{{(15 a^{2})}^{4}}{{(y^{3})}^{4}} {(5 a y^{- 2})}^{- 5}$

Apply the product rule to $15 a^{2}$ .

$\frac{15^{4} {(a^{2})}^{4}}{{(y^{3})}^{4}} {(5 a y^{- 2})}^{- 5}$

Step 5

Simplify the numerator.

Raise $15$ to the power of $4$ .

$\frac{50625 {(a^{2})}^{4}}{{(y^{3})}^{4}} {(5 a y^{- 2})}^{- 5}$

Multiply the exponents in ${(a^{2})}^{4}$ .

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

$\frac{50625 a^{2 \cdot 4}}{{(y^{3})}^{4}} {(5 a y^{- 2})}^{- 5}$

Multiply $2$ by $4$ .

$\frac{50625 a^{8}}{{(y^{3})}^{4}} {(5 a y^{- 2})}^{- 5}$

Step 6

Multiply the exponents in ${(y^{3})}^{4}$ .

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

$\frac{50625 a^{8}}{y^{3 \cdot 4}} {(5 a y^{- 2})}^{- 5}$

Multiply $3$ by $4$ .

$\frac{50625 a^{8}}{y^{12}} {(5 a y^{- 2})}^{- 5}$

Step 7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{50625 a^{8}}{y^{12}} {(5 a \frac{1}{y^{2}})}^{- 5}$

Step 8

Multiply $5 a \frac{1}{y^{2}}$ .

Combine $5$ and $\frac{1}{y^{2}}$ .

$\frac{50625 a^{8}}{y^{12}} {(a \frac{5}{y^{2}})}^{- 5}$

Combine $a$ and $\frac{5}{y^{2}}$ .

$\frac{50625 a^{8}}{y^{12}} {(\frac{a \cdot 5}{y^{2}})}^{- 5}$

Step 9

Move $5$ to the left of $a$ .

$\frac{50625 a^{8}}{y^{12}} {(\frac{5 \cdot a}{y^{2}})}^{- 5}$

Step 10

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{50625 a^{8}}{y^{12}} {(\frac{y^{2}}{5 a})}^{5}$

Step 11

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{y^{2}}{5 a}$ .

$\frac{50625 a^{8}}{y^{12}} \cdot \frac{{(y^{2})}^{5}}{{(5 a)}^{5}}$

Apply the product rule to $5 a$ .

$\frac{50625 a^{8}}{y^{12}} \cdot \frac{{(y^{2})}^{5}}{5^{5} a^{5}}$

Step 12

Combine.

$\frac{50625 a^{8} {(y^{2})}^{5}}{y^{12} (5^{5} a^{5})}$

Step 13

Cancel the common factor of $a^{8}$ and $a^{5}$ .

Factor $a^{5}$ out of $50625 a^{8} {(y^{2})}^{5}$ .

$\frac{a^{5} (50625 a^{3} {(y^{2})}^{5})}{y^{12} (5^{5} a^{5})}$

Cancel the common factors.

Factor $a^{5}$ out of $y^{12} (5^{5} a^{5})$ .

$\frac{a^{5} (50625 a^{3} {(y^{2})}^{5})}{a^{5} (y^{12} \cdot (5^{5}))}$

Cancel the common factor.

$\frac{a^{5} (50625 a^{3} {(y^{2})}^{5})}{a^{5} (y^{12} \cdot (5^{5}))}$

Rewrite the expression.

$\frac{50625 a^{3} {(y^{2})}^{5}}{y^{12} \cdot (5^{5})}$

Step 14

Simplify the expression.

Multiply the exponents in ${(y^{2})}^{5}$ .

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

$\frac{50625 a^{3} y^{2 \cdot 5}}{y^{12} \cdot 5^{5}}$

Multiply $2$ by $5$ .

$\frac{50625 a^{3} y^{10}}{y^{12} \cdot 5^{5}}$

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

$\frac{50625 a^{3} y^{10}}{y^{12} \cdot 3125}$

Step 15

Cancel the common factor of $50625$ and $3125$ .

Factor $625$ out of $50625 a^{3} y^{10}$ .

$\frac{625 (81 a^{3} y^{10})}{y^{12} \cdot 3125}$

Cancel the common factors.

Factor $625$ out of $y^{12} \cdot 3125$ .

$\frac{625 (81 a^{3} y^{10})}{625 (y^{12} \cdot 5)}$

Cancel the common factor.

$\frac{625 (81 a^{3} y^{10})}{625 (y^{12} \cdot 5)}$

Rewrite the expression.

$\frac{81 a^{3} y^{10}}{y^{12} \cdot 5}$

Step 16

Cancel the common factor of $y^{10}$ and $y^{12}$ .

Factor $y^{10}$ out of $81 a^{3} y^{10}$ .

$\frac{y^{10} (81 a^{3})}{y^{12} \cdot 5}$

Cancel the common factors.

Factor $y^{10}$ out of $y^{12} \cdot 5$ .

$\frac{y^{10} (81 a^{3})}{y^{10} (y^{2} \cdot 5)}$

Cancel the common factor.

$\frac{y^{10} (81 a^{3})}{y^{10} (y^{2} \cdot 5)}$

Rewrite the expression.

$\frac{81 a^{3}}{y^{2} \cdot 5}$

Step 17

Move $5$ to the left of $y^{2}$ .

$\frac{81 a^{3}}{5 y^{2}}$

Simplify (15a^2y^-3)^4(5ay^-2)^-5

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