Simplify (20t^-6a^-3)/((4t^-3a^2)^-2)

$\frac{20 t^{- 6} a^{- 3}}{{(4 t^{- 3} a^{2})}^{- 2}}$

Step 1

Simplify.

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

$\frac{20 a^{- 3}}{{(4 t^{- 3} a^{2})}^{- 2} t^{6}}$

Move $a^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{20}{{(4 t^{- 3} a^{2})}^{- 2} t^{6} a^{3}}$

Move ${(4 t^{- 3} a^{2})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{20 {(4 t^{- 3} a^{2})}^{2}}{t^{6} a^{3}}$

Step 2

Simplify the numerator.

Apply the product rule to $4 t^{- 3} a^{2}$ .

$\frac{20 {(4 t^{- 3})}^{2} {(a^{2})}^{2}}{t^{6} a^{3}}$

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

$\frac{20 {(4 \frac{1}{t^{3}})}^{2} {(a^{2})}^{2}}{t^{6} a^{3}}$

Combine $4$ and $\frac{1}{t^{3}}$ .

$\frac{20 {(\frac{4}{t^{3}})}^{2} {(a^{2})}^{2}}{t^{6} a^{3}}$

Apply the product rule to $\frac{4}{t^{3}}$ .

$\frac{20 \frac{4^{2}}{{(t^{3})}^{2}} {(a^{2})}^{2}}{t^{6} a^{3}}$

Raise $4$ to the power of $2$ .

$\frac{20 \frac{16}{{(t^{3})}^{2}} {(a^{2})}^{2}}{t^{6} a^{3}}$

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

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

$\frac{20 \frac{16}{t^{3 \cdot 2}} {(a^{2})}^{2}}{t^{6} a^{3}}$

Multiply $3$ by $2$ .

$\frac{20 \frac{16}{t^{6}} {(a^{2})}^{2}}{t^{6} a^{3}}$

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

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

$\frac{20 \frac{16}{t^{6}} a^{2 \cdot 2}}{t^{6} a^{3}}$

Multiply $2$ by $2$ .

$\frac{20 \frac{16}{t^{6}} a^{4}}{t^{6} a^{3}}$

Combine exponents.

Combine $20$ and $\frac{16}{t^{6}}$ .

$\frac{\frac{20 \cdot 16}{t^{6}} a^{4}}{t^{6} a^{3}}$

Multiply $20$ by $16$ .

$\frac{\frac{320}{t^{6}} a^{4}}{t^{6} a^{3}}$

Combine $\frac{320}{t^{6}}$ and $a^{4}$ .

$\frac{\frac{320 a^{4}}{t^{6}}}{t^{6} a^{3}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{320 a^{4}}{t^{6}} \cdot \frac{1}{t^{6} a^{3}}$

Step 4

Combine.

$\frac{320 a^{4} \cdot 1}{t^{6} (t^{6} a^{3})}$

Step 5

Multiply $t^{6}$ by $t^{6}$ by adding the exponents.

Move $t^{6}$ .

$\frac{320 a^{4} \cdot 1}{t^{6} t^{6} a^{3}}$

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

$\frac{320 a^{4} \cdot 1}{t^{6 + 6} a^{3}}$

Add $6$ and $6$ .

$\frac{320 a^{4} \cdot 1}{t^{12} a^{3}}$

Step 6

Multiply $320 a^{4}$ by $1$ .

$\frac{320 a^{4}}{t^{12} a^{3}}$

Step 7

Cancel the common factor of $a^{4}$ and $a^{3}$ .

Factor $a^{3}$ out of $320 a^{4}$ .

$\frac{a^{3} (320 a)}{t^{12} a^{3}}$

Cancel the common factors.

Factor $a^{3}$ out of $t^{12} a^{3}$ .

$\frac{a^{3} (320 a)}{a^{3} t^{12}}$

Cancel the common factor.

$\frac{a^{3} (320 a)}{a^{3} t^{12}}$

Rewrite the expression.

$\frac{320 a}{t^{12}}$

Simplify (20t^-6a^-3)/((4t^-3a^2)^-2)

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