Simplify ((15t^-3)/(t^5))^-5(7/(t^5))

${(\frac{15 t^{- 3}}{t^{5}})}^{- 5} (\frac{7}{t^{5}})$

Step 1

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

${(\frac{15}{t^{5} t^{3}})}^{- 5} \frac{7}{t^{5}}$

Step 2

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

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

${(\frac{15}{t^{5 + 3}})}^{- 5} \frac{7}{t^{5}}$

Add $5$ and $3$ .

${(\frac{15}{t^{8}})}^{- 5} \frac{7}{t^{5}}$

Step 3

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

${(\frac{t^{8}}{15})}^{5} \frac{7}{t^{5}}$

Step 4

Apply the product rule to $\frac{t^{8}}{15}$ .

$\frac{{(t^{8})}^{5}}{15^{5}} \cdot \frac{7}{t^{5}}$

Step 5

Combine.

$\frac{{(t^{8})}^{5} \cdot 7}{15^{5} t^{5}}$

Step 6

Simplify the expression.

Multiply the exponents in ${(t^{8})}^{5}$ .

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

$\frac{t^{8 \cdot 5} \cdot 7}{15^{5} t^{5}}$

Multiply $8$ by $5$ .

$\frac{t^{40} \cdot 7}{15^{5} t^{5}}$

Raise $15$ to the power of $5$ .

$\frac{t^{40} \cdot 7}{759375 t^{5}}$

Step 7

Cancel the common factor of $t^{40}$ and $t^{5}$ .

Factor $t^{5}$ out of $t^{40} \cdot 7$ .

$\frac{t^{5} (t^{35} \cdot 7)}{759375 t^{5}}$

Cancel the common factors.

Factor $t^{5}$ out of $759375 t^{5}$ .

$\frac{t^{5} (t^{35} \cdot 7)}{t^{5} \cdot 759375}$

Cancel the common factor.

$\frac{t^{5} (t^{35} \cdot 7)}{t^{5} \cdot 759375}$

Rewrite the expression.

$\frac{t^{35} \cdot 7}{759375}$

Step 8

Move $7$ to the left of $t^{35}$ .

$\frac{7 t^{35}}{759375}$