Algebra Examples

$\frac{\frac{1}{27}}{3^{2 k}} = 27$

Step 1

Multiply both sides by $3^{2 k}$ .

$\frac{\frac{1}{27}}{3^{2 k}} \cdot 3^{2 k} = 27 \cdot 3^{2 k}$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $3^{2 k}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{\frac{1}{27}}{3^{2 k}} \cdot 3^{2 k} = 27 \cdot 3^{2 k}$

Step 2.1.1.2

Rewrite the expression.

$\frac{1}{27} = 27 \cdot 3^{2 k}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $27 \cdot 3^{2 k}$ .

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Step 2.2.1.1

Rewrite $27$ as $3^{3}$ .

$\frac{1}{27} = 3^{3} \cdot 3^{2 k}$

Step 2.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{27} = 3^{3 + 2 k}$

Step 3

Solve for $k$ .

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Step 3.1

Rewrite the equation as $3^{3 + 2 k} = \frac{1}{27}$ .

$3^{3 + 2 k} = \frac{1}{27}$

Step 3.2

Raise $27$ to the power of $1$ .

$3^{3 + 2 k} = \frac{1}{27^{1}}$

Step 3.3

Move $27^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$3^{3 + 2 k} = 27^{- 1}$

Step 3.4

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

$3^{3 + 2 k} = 3^{- 3}$

Step 3.5

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

$3 + 2 k = - 3$

Step 3.6

Solve for $k$ .

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Step 3.6.1

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

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Step 3.6.1.1

Subtract $3$ from both sides of the equation.

$2 k = - 3 - 3$

Step 3.6.1.2

Subtract $3$ from $- 3$ .

$2 k = - 6$

Step 3.6.2

Divide each term in $2 k = - 6$ by $2$ and simplify.

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Step 3.6.2.1

Divide each term in $2 k = - 6$ by $2$ .

$\frac{2 k}{2} = \frac{- 6}{2}$

Step 3.6.2.2

Simplify the left side.

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Step 3.6.2.2.1

Cancel the common factor of $2$ .

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Step 3.6.2.2.1.1

Cancel the common factor.

$\frac{2 k}{2} = \frac{- 6}{2}$

Step 3.6.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 6}{2}$

Step 3.6.2.3

Simplify the right side.

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Step 3.6.2.3.1

Divide $- 6$ by $2$ .

$k = - 3$