Simplify (15p^-1d^9)/(3(p^2d^3)^-3)

$\frac{15 p^{- 1} d^{9}}{3 {(p^{2} d^{3})}^{- 3}}$

Step 1

Reduce the expression by cancelling the common factors.

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

$\frac{15 d^{9}}{3 {(p^{2} d^{3})}^{- 3} p}$

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

$\frac{15 d^{9} {(p^{2} d^{3})}^{3}}{3 p}$

Cancel the common factor of $15$ and $3$ .

Factor $3$ out of $15 d^{9} {(p^{2} d^{3})}^{3}$ .

$\frac{3 (5 d^{9} {(p^{2} d^{3})}^{3})}{3 p}$

Cancel the common factors.

Factor $3$ out of $3 p$ .

$\frac{3 (5 d^{9} {(p^{2} d^{3})}^{3})}{3 (p)}$

Cancel the common factor.

$\frac{3 (5 d^{9} {(p^{2} d^{3})}^{3})}{3 p}$

Rewrite the expression.

$\frac{5 d^{9} {(p^{2} d^{3})}^{3}}{p}$

Step 2

Simplify the numerator.

Apply the product rule to $p^{2} d^{3}$ .

$\frac{5 d^{9} {(p^{2})}^{3} {(d^{3})}^{3}}{p}$

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

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

$\frac{5 d^{9} p^{2 \cdot 3} {(d^{3})}^{3}}{p}$

Multiply $2$ by $3$ .

$\frac{5 d^{9} p^{6} {(d^{3})}^{3}}{p}$

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

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

$\frac{5 d^{9} p^{6} d^{3 \cdot 3}}{p}$

Multiply $3$ by $3$ .

$\frac{5 d^{9} p^{6} d^{9}}{p}$

Multiply $d^{9}$ by $d^{9}$ by adding the exponents.

Move $d^{9}$ .

$\frac{5 (d^{9} d^{9}) p^{6}}{p}$

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

$\frac{5 d^{9 + 9} p^{6}}{p}$

Add $9$ and $9$ .

$\frac{5 d^{18} p^{6}}{p}$

Step 3

Cancel the common factor of $p^{6}$ and $p$ .

Factor $p$ out of $5 d^{18} p^{6}$ .

$\frac{p (5 d^{18} p^{5})}{p}$

Cancel the common factors.

Raise $p$ to the power of $1$ .

$\frac{p (5 d^{18} p^{5})}{p^{1}}$

Factor $p$ out of $p^{1}$ .

$\frac{p (5 d^{18} p^{5})}{p \cdot 1}$

Cancel the common factor.

$\frac{p (5 d^{18} p^{5})}{p \cdot 1}$

Rewrite the expression.

$\frac{5 d^{18} p^{5}}{1}$

Divide $5 d^{18} p^{5}$ by $1$ .

$5 d^{18} p^{5}$

Simplify (15p^-1d^9)/(3(p^2d^3)^-3)

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