Precalculus Examples

$- 1296 = - 1 {(\sqrt{6})}^{n - 1}$

Step 1

Rewrite the equation as $- 1 {\sqrt{6}}^{n - 1} = - 1296$ .

$- 1 {\sqrt{6}}^{n - 1} = - 1296$

Step 2

Divide each term in $- 1 {\sqrt{6}}^{n - 1} = - 1296$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $- 1 {\sqrt{6}}^{n - 1} = - 1296$ by $- 1$ .

$\frac{- 1 {\sqrt{6}}^{n - 1}}{- 1} = \frac{- 1296}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Dividing two negative values results in a positive value.

$\frac{{\sqrt{6}}^{n - 1}}{1} = \frac{- 1296}{- 1}$

Step 2.2.2

Divide ${\sqrt{6}}^{n - 1}$ by $1$ .

${\sqrt{6}}^{n - 1} = \frac{- 1296}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 1296$ by $- 1$ .

${\sqrt{6}}^{n - 1} = 1296$

Step 3

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

${(6^{\frac{1}{2}})}^{n - 1} = 1296$

Step 4

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

$6^{\frac{1}{2} (n - 1)} = 1296$

Step 5

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

$6^{\frac{1}{2} (n - 1)} = 6^{4}$

Step 6

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

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

Step 7

Solve for $n$ .

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Step 7.1

Multiply both sides of the equation by $2$ .

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

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Simplify $2 (\frac{1}{2} \cdot (n - 1))$ .

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Step 7.2.1.1.1

Apply the distributive property.

$2 (\frac{1}{2} n + \frac{1}{2} \cdot - 1) = 2 \cdot 4$

Step 7.2.1.1.2

Combine $\frac{1}{2}$ and $n$ .

$2 (\frac{n}{2} + \frac{1}{2} \cdot - 1) = 2 \cdot 4$

Step 7.2.1.1.3

Combine $\frac{1}{2}$ and $- 1$ .

$2 (\frac{n}{2} + \frac{- 1}{2}) = 2 \cdot 4$

Step 7.2.1.1.4

Move the negative in front of the fraction.

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

Step 7.2.1.1.5

Apply the distributive property.

$2 \frac{n}{2} + 2 (- \frac{1}{2}) = 2 \cdot 4$

Step 7.2.1.1.6

Cancel the common factor of $2$ .

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Step 7.2.1.1.6.1

Cancel the common factor.

$2 \frac{n}{2} + 2 (- \frac{1}{2}) = 2 \cdot 4$

Step 7.2.1.1.6.2

Rewrite the expression.

$n + 2 (- \frac{1}{2}) = 2 \cdot 4$

Step 7.2.1.1.7

Cancel the common factor of $2$ .

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Step 7.2.1.1.7.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$n + 2 (\frac{- 1}{2}) = 2 \cdot 4$

Step 7.2.1.1.7.2

Cancel the common factor.

$n + 2 (\frac{- 1}{2}) = 2 \cdot 4$

Step 7.2.1.1.7.3

Rewrite the expression.

$n - 1 = 2 \cdot 4$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Multiply $2$ by $4$ .

$n - 1 = 8$

Step 7.3

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

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Step 7.3.1

Add $1$ to both sides of the equation.

$n = 8 + 1$

Step 7.3.2

Add $8$ and $1$ .

$n = 9$