Simplify ((5t^-3)/(t^2))^-2*2/(t^-2)

${(\frac{5 t^{- 3}}{t^{2}})}^{- 2} \cdot \frac{2}{t^{- 2}}$

Step 1

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

${(\frac{5 t^{- 3}}{t^{2}})}^{- 2} \cdot \frac{2}{\frac{1}{t^{2}}}$

Step 2

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

${(\frac{5}{t^{2} t^{3}})}^{- 2} \cdot \frac{2}{\frac{1}{t^{2}}}$

Step 3

Multiply $t^{2}$ by $t^{3}$ by adding the exponents.

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

${(\frac{5}{t^{2 + 3}})}^{- 2} \cdot \frac{2}{\frac{1}{t^{2}}}$

Add $2$ and $3$ .

${(\frac{5}{t^{5}})}^{- 2} \cdot \frac{2}{\frac{1}{t^{2}}}$

Step 4

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

${(\frac{t^{5}}{5})}^{2} \cdot \frac{2}{\frac{1}{t^{2}}}$

Step 5

Apply the product rule to $\frac{t^{5}}{5}$ .

$\frac{{(t^{5})}^{2}}{5^{2}} \cdot \frac{2}{\frac{1}{t^{2}}}$

Step 6

Combine.

$\frac{{(t^{5})}^{2} \cdot 2}{5^{2} \frac{1}{t^{2}}}$

Step 7

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

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

$\frac{t^{5 \cdot 2} \cdot 2}{5^{2} \frac{1}{t^{2}}}$

Multiply $5$ by $2$ .

$\frac{t^{10} \cdot 2}{5^{2} \frac{1}{t^{2}}}$

Step 8

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

$\frac{t^{10} \cdot 2}{\frac{5^{2}}{t^{2}}}$

Step 9

Raise $5$ to the power of $2$ .

$\frac{t^{10} \cdot 2}{\frac{25}{t^{2}}}$

Step 10

Multiply the numerator by the reciprocal of the denominator.

$t^{10} \cdot 2 \frac{t^{2}}{25}$

Step 11

Move $2$ to the left of $t^{10}$ .

$2 \cdot t^{10} \frac{t^{2}}{25}$

Step 12

Multiply $2 t^{10} \frac{t^{2}}{25}$ .

Combine $2$ and $\frac{t^{2}}{25}$ .

$t^{10} \frac{2 t^{2}}{25}$

Combine $t^{10}$ and $\frac{2 t^{2}}{25}$ .

$\frac{t^{10} (2 t^{2})}{25}$

Multiply $t^{10}$ by $t^{2}$ by adding the exponents.

Move $t^{2}$ .

$\frac{t^{2} t^{10} \cdot 2}{25}$

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

$\frac{t^{2 + 10} \cdot 2}{25}$

Add $2$ and $10$ .

$\frac{t^{12} \cdot 2}{25}$

Step 13

Move $2$ to the left of $t^{12}$ .

$\frac{2 t^{12}}{25}$

Simplify ((5t^-3)/(t^2))^-2*2/(t^-2)

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