Algebra Examples

${(3^{2})}^{\sqrt{x - 1}} = 81$

Step 1

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

${(3^{2})}^{{(x - 1)}^{\frac{1}{2}}} = 81$

Step 2

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

$3^{2 {(x - 1)}^{\frac{1}{2}}} = 81$

Step 3

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

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

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

Divide each term in $2 {(x - 1)}^{\frac{1}{2}} = 4$ by $2$ and simplify.

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Step 5.1.1

Divide each term in $2 {(x - 1)}^{\frac{1}{2}} = 4$ by $2$ .

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Cancel the common factor.

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

Step 5.1.2.2

Divide ${(x - 1)}^{\frac{1}{2}}$ by $1$ .

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

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Divide $4$ by $2$ .

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

Step 5.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 5.3

Simplify the exponent.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.1.1.1

Multiply the exponents in ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

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Step 5.3.1.1.1.1

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

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

Step 5.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 5.3.1.1.1.2.1

Cancel the common factor.

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

Step 5.3.1.1.1.2.2

Rewrite the expression.

${(x - 1)}^{1} = 2^{2}$

Step 5.3.1.1.2

Simplify.

$x - 1 = 2^{2}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Raise $2$ to the power of $2$ .

$x - 1 = 4$

Step 5.4

Move all terms not containing $x$ to the right side of the equation.

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Step 5.4.1

Add $1$ to both sides of the equation.

$x = 4 + 1$

Step 5.4.2

Add $4$ and $1$ .

$x = 5$