Solve for k ( square root of 3)^(10k)=81^(k+3)

${(\sqrt{3})}^{10 k} = 81^{k + 3}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

${(3^{\frac{1}{2}})}^{10 k} = 81^{k + 3}$

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

$3^{\frac{1}{2} (10 k)} = 81^{k + 3}$

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

$3^{\frac{1}{2} (10 k)} = 3^{4 (k + 3)}$

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

$\frac{1}{2} (10 k) = 4 (k + 3)$

Solve for $k$ .

Simplify $\frac{1}{2} \cdot (10 k)$ .

Start simplifying.

$0 + 0 + \frac{1}{2} \cdot (10 k) = 4 (k + 3)$

Simplify by adding zeros.

$\frac{1}{2} \cdot (10 k) = 4 (k + 3)$

Cancel the common factor of $2$ .

Factor $2$ out of $10 k$ .

$\frac{1}{2} \cdot (2 (5 k)) = 4 (k + 3)$

Cancel the common factor.

$\frac{1}{2} \cdot (2 (5 k)) = 4 (k + 3)$

Rewrite the expression.

$5 k = 4 (k + 3)$

Simplify $4 (k + 3)$ .

Apply the distributive property.

$5 k = 4 k + 4 \cdot 3$

Multiply $4$ by $3$ .

$5 k = 4 k + 12$

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

Subtract $4 k$ from both sides of the equation.

$5 k - 4 k = 12$

Subtract $4 k$ from $5 k$ .

$k = 12$