Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Solve for $k$ .

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Step 5.1

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

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Step 5.1.1

Rewrite.

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

Step 5.1.2

Simplify by adding zeros.

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

Step 5.1.3

Cancel the common factor of $2$ .

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Step 5.1.3.1

Factor $2$ out of $10 k$ .

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

Step 5.1.3.2

Cancel the common factor.

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

Step 5.1.3.3

Rewrite the expression.

$5 k = 4 (k + 3)$

Step 5.2

Simplify $4 (k + 3)$ .

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Step 5.2.1

Apply the distributive property.

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

Step 5.2.2

Multiply $4$ by $3$ .

$5 k = 4 k + 12$

Step 5.3

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

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Step 5.3.1

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

$5 k - 4 k = 12$

Step 5.3.2

Subtract $4 k$ from $5 k$ .

$k = 12$