Algebra Examples

${(\sqrt{7})}^{6 x} = 49^{x - 6}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

${(7^{\frac{1}{2}})}^{6 x} = 49^{x - 6}$

Step 2

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

$7^{\frac{1}{2} (6 x)} = 49^{x - 6}$

Step 3

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

$7^{\frac{1}{2} (6 x)} = 7^{2 (x - 6)}$

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

Simplify $\frac{1}{2} \cdot (6 x)$ .

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Step 5.1.1

Rewrite.

$0 + 0 + \frac{1}{2} \cdot (6 x) = 2 (x - 6)$

Step 5.1.2

Simplify by adding zeros.

$\frac{1}{2} \cdot (6 x) = 2 (x - 6)$

Step 5.1.3

Cancel the common factor of $2$ .

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Step 5.1.3.1

Factor $2$ out of $6 x$ .

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

Step 5.1.3.2

Cancel the common factor.

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

Step 5.1.3.3

Rewrite the expression.

$3 x = 2 (x - 6)$

Step 5.2

Simplify $2 (x - 6)$ .

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Step 5.2.1

Apply the distributive property.

$3 x = 2 x + 2 \cdot - 6$

Step 5.2.2

Multiply $2$ by $- 6$ .

$3 x = 2 x - 12$

Step 5.3

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

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Step 5.3.1

Subtract $2 x$ from both sides of the equation.

$3 x - 2 x = - 12$

Step 5.3.2

Subtract $2 x$ from $3 x$ .

$x = - 12$