Solve for a 9^(-4a)=243^(2a+3)

$9^{- 4 a} = 243^{2 a + 3}$

Step 1

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

$3^{2 (- 4 a)} = 3^{5 (2 a + 3)}$

Step 2

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

$2 (- 4 a) = 5 (2 a + 3)$

Step 3

Solve for $a$ .

Multiply $- 4$ by $2$ .

$- 8 a = 5 (2 a + 3)$

Simplify $5 (2 a + 3)$ .

Apply the distributive property.

$- 8 a = 5 (2 a) + 5 \cdot 3$

Multiply.

Multiply $2$ by $5$ .

$- 8 a = 10 a + 5 \cdot 3$

Multiply $5$ by $3$ .

$- 8 a = 10 a + 15$

Move all terms containing $a$ to the left side of the equation.

Subtract $10 a$ from both sides of the equation.

$- 8 a - 10 a = 15$

Subtract $10 a$ from $- 8 a$ .

$- 18 a = 15$

Divide each term in $- 18 a = 15$ by $- 18$ and simplify.

Divide each term in $- 18 a = 15$ by $- 18$ .

$\frac{- 18 a}{- 18} = \frac{15}{- 18}$

Simplify the left side.

Cancel the common factor of $- 18$ .

Cancel the common factor.

$\frac{- 18 a}{- 18} = \frac{15}{- 18}$

Divide $a$ by $1$ .

$a = \frac{15}{- 18}$

Simplify the right side.

Cancel the common factor of $15$ and $- 18$ .

Factor $3$ out of $15$ .

$a = \frac{3 (5)}{- 18}$

Cancel the common factors.

Factor $3$ out of $- 18$ .

$a = \frac{3 \cdot 5}{3 \cdot - 6}$

Cancel the common factor.

$a = \frac{3 \cdot 5}{3 \cdot - 6}$

Rewrite the expression.

$a = \frac{5}{- 6}$

Move the negative in front of the fraction.

$a = - \frac{5}{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{5}{6}$

Decimal Form:

$a = - 0.8\overline{3}$

Solve for a 9^(-4a)=243^(2a+3)

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