Solve for m 5^m*(5^-7)^m=5^12

$5^{m} \cdot {(5^{- 7})}^{m} = 5^{12}$

Step 1

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

$5^{m} \cdot 5^{- 7 m} = 5^{12}$

Step 2

Multiply $5^{m}$ by $5^{- 7 m}$ by adding the exponents.

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

$5^{m - 7 m} = 5^{12}$

Subtract $7 m$ from $m$ .

$5^{- 6 m} = 5^{12}$

Step 3

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

$- 6 m = 12$

Step 4

Divide each term in $- 6 m = 12$ by $- 6$ and simplify.

Divide each term in $- 6 m = 12$ by $- 6$ .

$\frac{- 6 m}{- 6} = \frac{12}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 m}{- 6} = \frac{12}{- 6}$

Divide $m$ by $1$ .

$m = \frac{12}{- 6}$

Simplify the right side.

Divide $12$ by $- 6$ .

$m = - 2$