Solve for z 5^8*5^z=1

$5^{8} \cdot 5^{z} = 1$

Step 1

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

$5^{8 + z} = 1$

Step 2

Divide each term in $5^{8 + z} = 1$ by $5^{8}$ and simplify.

Divide each term in $5^{8 + z} = 1$ by $5^{8}$ .

$\frac{5^{8 + z}}{5^{8}} = \frac{1}{5^{8}}$

Simplify the left side.

Cancel the common factor of $5^{8 + z}$ and $5^{8}$ .

Factor $5^{8}$ out of $5^{8 + z}$ .

$\frac{5^{8} \cdot 5^{z}}{5^{8}} = \frac{1}{5^{8}}$

Cancel the common factors.

Multiply by $1$ .

$\frac{5^{8} \cdot 5^{z}}{5^{8} \cdot 1} = \frac{1}{5^{8}}$

Cancel the common factor.

$\frac{5^{8} \cdot 5^{z}}{5^{8} \cdot 1} = \frac{1}{5^{8}}$

Rewrite the expression.

$\frac{5^{z}}{1} = \frac{1}{5^{8}}$

Divide $5^{z}$ by $1$ .

$5^{z} = \frac{1}{5^{8}}$

Simplify the right side.

Raise $5$ to the power of $8$ .

$5^{z} = \frac{1}{390625}$

Step 3

Raise $390625$ to the power of $1$ .

$5^{z} = \frac{1}{390625^{1}}$

Step 4

Move $390625^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$5^{z} = 390625^{- 1}$

Step 5

Create equivalent expressions in the equation that all have equal bases.

$5^{z} = 5^{- 8}$

Step 6

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

$z = - 8$