Solve for n p^3q^2((p^4q^n*r^3)/(r^-4))=p^7q^5r^7

$p^{3} \cdot q^{2} (\frac{p^{4} \cdot q^{n} \cdot r^{3}}{r^{- 4}}) = p^{7} q^{5} r^{7}$

Step 1

Simplify.

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

$p^{3} \cdot (q^{2} (p^{4} \cdot q^{n} \cdot r^{3} r^{4})) = p^{7} q^{5} r^{7}$

Multiply $r^{3}$ by $r^{4}$ by adding the exponents.

Move $r^{4}$ .

$p^{3} \cdot (q^{2} (p^{4} \cdot q^{n} \cdot (r^{4} r^{3}))) = p^{7} q^{5} r^{7}$

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

$p^{3} \cdot (q^{2} (p^{4} \cdot q^{n} \cdot r^{4 + 3})) = p^{7} q^{5} r^{7}$

Add $4$ and $3$ .

$p^{3} \cdot (q^{2} (p^{4} \cdot q^{n} \cdot r^{7})) = p^{7} q^{5} r^{7}$

Multiply $q^{2}$ by $q^{n}$ by adding the exponents.

Move $q^{n}$ .

$p^{3} \cdot (q^{n} q^{2} (p^{4} \cdot r^{7})) = p^{7} q^{5} r^{7}$

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

$p^{3} \cdot (q^{n + 2} (p^{4} \cdot r^{7})) = p^{7} q^{5} r^{7}$

Multiply $p^{3}$ by $p^{4}$ by adding the exponents.

Move $p^{4}$ .

$p^{4} p^{3} \cdot (q^{n + 2} (r^{7})) = p^{7} q^{5} r^{7}$

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

$p^{4 + 3} \cdot (q^{n + 2} (r^{7})) = p^{7} q^{5} r^{7}$

Add $4$ and $3$ .

$p^{7} \cdot (q^{n + 2} (r^{7})) = p^{7} q^{5} r^{7}$

Multiply $q^{n + 2}$ by $r^{7}$ .

$p^{7} \cdot (q^{n + 2} r^{7}) = p^{7} q^{5} r^{7}$

Remove parentheses.

$p^{7} \cdot q^{n + 2} r^{7} = p^{7} q^{5} r^{7}$

Step 2

Divide each term in $(p^{7} \cdot (q^{n + 2} r^{7})) = p^{7} q^{5} r^{7}$ by $p^{3} q^{2}$ and simplify.

Divide each term in $(p^{7} \cdot (q^{n + 2} r^{7})) = p^{7} q^{5} r^{7}$ by $p^{3} q^{2}$ .

$\frac{(p^{7} \cdot (q^{n + 2} r^{7}))}{p^{3} q^{2}} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Simplify the left side.

Cancel the common factor of $p^{7}$ and $p^{3}$ .

Factor $p^{3}$ out of $p^{7} \cdot q^{n + 2} r^{7}$ .

$\frac{p^{3} (p^{4} \cdot q^{n + 2} r^{7})}{p^{3} q^{2}} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Cancel the common factors.

Factor $p^{3}$ out of $p^{3} q^{2}$ .

$\frac{p^{3} (p^{4} \cdot q^{n + 2} r^{7})}{p^{3} (q^{2})} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Cancel the common factor.

$\frac{p^{3} (p^{4} \cdot q^{n + 2} r^{7})}{p^{3} q^{2}} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Rewrite the expression.

$\frac{p^{4} \cdot q^{n + 2} r^{7}}{q^{2}} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Cancel the common factor of $q^{n + 2}$ and $q^{2}$ .

Factor $q^{2}$ out of $p^{4} \cdot q^{n + 2} r^{7}$ .

$\frac{q^{2} (p^{4} \cdot q^{n} r^{7})}{q^{2}} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Cancel the common factors.

Multiply by $1$ .

$\frac{q^{2} (p^{4} \cdot q^{n} r^{7})}{q^{2} \cdot 1} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Cancel the common factor.

$\frac{q^{2} (p^{4} \cdot q^{n} r^{7})}{q^{2} \cdot 1} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Rewrite the expression.

$\frac{p^{4} \cdot q^{n} r^{7}}{1} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Divide $p^{4} \cdot q^{n} r^{7}$ by $1$ .

$p^{4} \cdot q^{n} r^{7} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Reorder factors in $p^{4} q^{n} r^{7}$ .

$p^{4} r^{7} q^{n} = \frac{p^{7} q^{5} r^{7}}{p^{3} q^{2}}$

Simplify the right side.

Cancel the common factor of $p^{7}$ and $p^{3}$ .

Factor $p^{3}$ out of $p^{7} q^{5} r^{7}$ .

$p^{4} r^{7} q^{n} = \frac{p^{3} (p^{4} q^{5} r^{7})}{p^{3} q^{2}}$

Cancel the common factors.

Factor $p^{3}$ out of $p^{3} q^{2}$ .

$p^{4} r^{7} q^{n} = \frac{p^{3} (p^{4} q^{5} r^{7})}{p^{3} (q^{2})}$

Cancel the common factor.

$p^{4} r^{7} q^{n} = \frac{p^{3} (p^{4} q^{5} r^{7})}{p^{3} q^{2}}$

Rewrite the expression.

$p^{4} r^{7} q^{n} = \frac{p^{4} q^{5} r^{7}}{q^{2}}$

Cancel the common factor of $q^{5}$ and $q^{2}$ .

Factor $q^{2}$ out of $p^{4} q^{5} r^{7}$ .

$p^{4} r^{7} q^{n} = \frac{q^{2} (p^{4} q^{3} r^{7})}{q^{2}}$

Cancel the common factors.

Multiply by $1$ .

$p^{4} r^{7} q^{n} = \frac{q^{2} (p^{4} q^{3} r^{7})}{q^{2} \cdot 1}$

Cancel the common factor.

$p^{4} r^{7} q^{n} = \frac{q^{2} (p^{4} q^{3} r^{7})}{q^{2} \cdot 1}$

Rewrite the expression.

$p^{4} r^{7} q^{n} = \frac{p^{4} q^{3} r^{7}}{1}$

Divide $p^{4} q^{3} r^{7}$ by $1$ .

$p^{4} r^{7} q^{n} = p^{4} q^{3} r^{7}$

Step 3

Divide each term in $p^{4} r^{7} q^{n} = p^{4} q^{3} r^{7}$ by $p^{4} r^{7}$ and simplify.

Divide each term in $p^{4} r^{7} q^{n} = p^{4} q^{3} r^{7}$ by $p^{4} r^{7}$ .

$\frac{p^{4} r^{7} q^{n}}{p^{4} r^{7}} = \frac{p^{4} q^{3} r^{7}}{p^{4} r^{7}}$

Simplify the left side.

Cancel the common factor of $p^{4}$ .

Cancel the common factor.

$\frac{p^{4} r^{7} q^{n}}{p^{4} r^{7}} = \frac{p^{4} q^{3} r^{7}}{p^{4} r^{7}}$

Rewrite the expression.

$\frac{r^{7} q^{n}}{r^{7}} = \frac{p^{4} q^{3} r^{7}}{p^{4} r^{7}}$

Cancel the common factor of $r^{7}$ .

Cancel the common factor.

$\frac{r^{7} q^{n}}{r^{7}} = \frac{p^{4} q^{3} r^{7}}{p^{4} r^{7}}$

Divide $q^{n}$ by $1$ .

$q^{n} = \frac{p^{4} q^{3} r^{7}}{p^{4} r^{7}}$

Simplify the right side.

Cancel the common factor of $p^{4}$ .

Cancel the common factor.

$q^{n} = \frac{p^{4} q^{3} r^{7}}{p^{4} r^{7}}$

Rewrite the expression.

$q^{n} = \frac{q^{3} r^{7}}{r^{7}}$

Cancel the common factor of $r^{7}$ .

Cancel the common factor.

$q^{n} = \frac{q^{3} r^{7}}{r^{7}}$

Divide $q^{3}$ by $1$ .

$q^{n} = q^{3}$

Step 4

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$n = 3$

Solve for n p^3*q^2((p^4*q^n*r^3)/(r^-4))=p^7q^5r^7

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Solve for n p^3q^2((p^4q^n*r^3)/(r^-4))=p^7q^5r^7