Simplify (4z^-4t^10)/((3z^3t^-4)^2)

$\frac{4 z^{- 4} t^{10}}{{(3 z^{3} t^{- 4})}^{2}}$

Step 1

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

$\frac{4 t^{10}}{{(3 z^{3} t^{- 4})}^{2} z^{4}}$

Step 2

Simplify the denominator.

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

$\frac{4 t^{10}}{{(3 z^{3})}^{2} {(t^{- 4})}^{2} z^{4}}$

Apply the product rule to $3 z^{3}$ .

$\frac{4 t^{10}}{3^{2} {(z^{3})}^{2} {(t^{- 4})}^{2} z^{4}}$

Raise $3$ to the power of $2$ .

$\frac{4 t^{10}}{9 {(z^{3})}^{2} {(t^{- 4})}^{2} z^{4}}$

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

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

$\frac{4 t^{10}}{9 z^{3 \cdot 2} {(t^{- 4})}^{2} z^{4}}$

Multiply $3$ by $2$ .

$\frac{4 t^{10}}{9 z^{6} {(t^{- 4})}^{2} z^{4}}$

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

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

$\frac{4 t^{10}}{9 z^{6} t^{- 4 \cdot 2} z^{4}}$

Multiply $- 4$ by $2$ .

$\frac{4 t^{10}}{9 z^{6} t^{- 8} z^{4}}$

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

$\frac{4 t^{10}}{9 z^{6} \frac{1}{t^{8}} z^{4}}$

Combine exponents.

Combine $9$ and $\frac{1}{t^{8}}$ .

$\frac{4 t^{10}}{z^{6} \frac{9}{t^{8}} z^{4}}$

Combine $z^{6}$ and $\frac{9}{t^{8}}$ .

$\frac{4 t^{10}}{\frac{z^{6} \cdot 9}{t^{8}} z^{4}}$

Combine $\frac{z^{6} \cdot 9}{t^{8}}$ and $z^{4}$ .

$\frac{4 t^{10}}{\frac{z^{6} \cdot 9 z^{4}}{t^{8}}}$

Multiply $z^{6}$ by $z^{4}$ by adding the exponents.

Move $z^{4}$ .

$\frac{4 t^{10}}{\frac{z^{4} z^{6} \cdot 9}{t^{8}}}$

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

$\frac{4 t^{10}}{\frac{z^{4 + 6} \cdot 9}{t^{8}}}$

Add $4$ and $6$ .

$\frac{4 t^{10}}{\frac{z^{10} \cdot 9}{t^{8}}}$

Move $9$ to the left of $z^{10}$ .

$\frac{4 t^{10}}{\frac{9 z^{10}}{t^{8}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$4 t^{10} \frac{t^{8}}{9 z^{10}}$

Step 4

Multiply $4 t^{10} \frac{t^{8}}{9 z^{10}}$ .

Combine $4$ and $\frac{t^{8}}{9 z^{10}}$ .

$t^{10} \frac{4 t^{8}}{9 z^{10}}$

Combine $t^{10}$ and $\frac{4 t^{8}}{9 z^{10}}$ .

$\frac{t^{10} (4 t^{8})}{9 z^{10}}$

Multiply $t^{10}$ by $t^{8}$ by adding the exponents.

Move $t^{8}$ .

$\frac{t^{8} t^{10} \cdot 4}{9 z^{10}}$

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

$\frac{t^{8 + 10} \cdot 4}{9 z^{10}}$

Add $8$ and $10$ .

$\frac{t^{18} \cdot 4}{9 z^{10}}$

Step 5

Move $4$ to the left of $t^{18}$ .

$\frac{4 t^{18}}{9 z^{10}}$

Simplify (4z^-4t^10)/((3z^3t^-4)^2)

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