Solve for x Solve ( square root of 6)^(8x)=216^(x-3)

Solve ${(\sqrt{6})}^{8 x} = 216^{x - 3}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

${(6^{\frac{1}{2}})}^{8 x} = 216^{x - 3}$

Step 2

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

$6^{\frac{1}{2} (8 x)} = 216^{x - 3}$

Step 3

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

$6^{\frac{1}{2} (8 x)} = 6^{3 (x - 3)}$

Step 4

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

$\frac{1}{2} (8 x) = 3 (x - 3)$

Step 5

Solve for $x$ .

Simplify $\frac{1}{2} \cdot (8 x)$ .

Start simplifying.

$0 + 0 + \frac{1}{2} \cdot (8 x) = 3 (x - 3)$

Simplify by adding zeros.

$\frac{1}{2} \cdot (8 x) = 3 (x - 3)$

Cancel the common factor of $2$ .

Factor $2$ out of $8 x$ .

$\frac{1}{2} \cdot (2 (4 x)) = 3 (x - 3)$

Cancel the common factor.

$\frac{1}{2} \cdot (2 (4 x)) = 3 (x - 3)$

Rewrite the expression.

$4 x = 3 (x - 3)$

Simplify $3 (x - 3)$ .

Apply the distributive property.

$4 x = 3 x + 3 \cdot - 3$

Multiply $3$ by $- 3$ .

$4 x = 3 x - 9$

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

Subtract $3 x$ from both sides of the equation.

$4 x - 3 x = - 9$

Subtract $3 x$ from $4 x$ .

$x = - 9$