Simplify 7t^11u^3(-4t^-9)^2u^-5

$7 t^{11} u^{3} {(- 4 t^{- 9})}^{2} u^{- 5}$

Step 1

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

Move $u^{- 5}$ .

$7 t^{11} (u^{- 5} u^{3}) {(- 4 t^{- 9})}^{2}$

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

$7 t^{11} u^{- 5 + 3} {(- 4 t^{- 9})}^{2}$

Add $- 5$ and $3$ .

$7 t^{11} u^{- 2} {(- 4 t^{- 9})}^{2}$

Step 2

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

$7 t^{11} \frac{1}{u^{2}} {(- 4 t^{- 9})}^{2}$

Step 3

Multiply $7 t^{11} \frac{1}{u^{2}}$ .

Combine $7$ and $\frac{1}{u^{2}}$ .

$t^{11} \frac{7}{u^{2}} {(- 4 t^{- 9})}^{2}$

Combine $t^{11}$ and $\frac{7}{u^{2}}$ .

$\frac{t^{11} \cdot 7}{u^{2}} {(- 4 t^{- 9})}^{2}$

Step 4

Move $7$ to the left of $t^{11}$ .

$\frac{7 \cdot t^{11}}{u^{2}} {(- 4 t^{- 9})}^{2}$

Step 5

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

$\frac{7 t^{11}}{u^{2}} {(- 4 \frac{1}{t^{9}})}^{2}$

Step 6

Combine $- 4$ and $\frac{1}{t^{9}}$ .

$\frac{7 t^{11}}{u^{2}} {(\frac{- 4}{t^{9}})}^{2}$

Step 7

Move the negative in front of the fraction.

$\frac{7 t^{11}}{u^{2}} {(- \frac{4}{t^{9}})}^{2}$

Step 8

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

Apply the product rule to $- \frac{4}{t^{9}}$ .

$\frac{7 t^{11}}{u^{2}} ({(- 1)}^{2} {(\frac{4}{t^{9}})}^{2})$

Apply the product rule to $\frac{4}{t^{9}}$ .

$\frac{7 t^{11}}{u^{2}} ({(- 1)}^{2} \frac{4^{2}}{{(t^{9})}^{2}})$

Step 9

Combine fractions.

Rewrite using the commutative property of multiplication.

${(- 1)}^{2} \frac{7 t^{11}}{u^{2}} \frac{4^{2}}{{(t^{9})}^{2}}$

Simplify the expression.

Raise $- 1$ to the power of $2$ .

$1 \frac{7 t^{11}}{u^{2}} \frac{4^{2}}{{(t^{9})}^{2}}$

Multiply $\frac{7 t^{11}}{u^{2}}$ by $1$ .

$\frac{7 t^{11}}{u^{2}} \cdot \frac{4^{2}}{{(t^{9})}^{2}}$

Combine.

$\frac{7 t^{11} \cdot 4^{2}}{u^{2} {(t^{9})}^{2}}$

Step 10

Simplify the numerator.

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

$\frac{7 t^{11} \cdot 16}{u^{2} {(t^{9})}^{2}}$

Multiply $16$ by $7$ .

$\frac{112 t^{11}}{u^{2} {(t^{9})}^{2}}$

Step 11

Reduce the expression by cancelling the common factors.

Multiply the exponents in ${(t^{9})}^{2}$ .

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

$\frac{112 t^{11}}{u^{2} t^{9 \cdot 2}}$

Multiply $9$ by $2$ .

$\frac{112 t^{11}}{u^{2} t^{18}}$

Cancel the common factor of $t^{11}$ and $t^{18}$ .

Factor $t^{11}$ out of $112 t^{11}$ .

$\frac{t^{11} \cdot 112}{u^{2} t^{18}}$

Cancel the common factors.

Factor $t^{11}$ out of $u^{2} t^{18}$ .

$\frac{t^{11} \cdot 112}{t^{11} (u^{2} t^{7})}$

Cancel the common factor.

$\frac{t^{11} \cdot 112}{t^{11} (u^{2} t^{7})}$

Rewrite the expression.

$\frac{112}{u^{2} t^{7}}$

Simplify 7t^11u^3(-4t^-9)^2u^-5

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