Solve for a (3^-1)÷(3^4)=3^a

$3^{- 1} \div 3^{4} = 3^{a}$

Step 1

Rewrite the equation as $3^{a} = 3^{- 1} \div 3^{4}$ .

$3^{a} = 3^{- 1} \div 3^{4}$

Step 2

Move $3^{4}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{a} = 3^{- 1} \cdot 3^{- 4}$

Step 3

Multiply $3^{- 1}$ by $3^{- 4}$ by adding the exponents.

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

$3^{a} = 3^{- 1 - 4}$

Subtract $4$ from $- 1$ .

$3^{a} = 3^{- 5}$

Step 4

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

$a = - 5$