Precalculus Examples

$243^{k + 2} \cdot 9^{2 k - 1} = 9$

Step 1

Rewrite $243$ as $3^{5}$ .

${(3^{5})}^{k + 2} \cdot 9^{2 k - 1} = 9$

Step 2

Multiply the exponents in ${(3^{5})}^{k + 2}$ .

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Step 2.1

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

$3^{5 (k + 2)} \cdot 9^{2 k - 1} = 9$

Step 2.2

Apply the distributive property.

$3^{5 k + 5 \cdot 2} \cdot 9^{2 k - 1} = 9$

Step 2.3

Multiply $5$ by $2$ .

$3^{5 k + 10} \cdot 9^{2 k - 1} = 9$

Step 3

Rewrite $9$ as $3^{2}$ .

$3^{5 k + 10} \cdot {(3^{2})}^{2 k - 1} = 9$

Step 4

Multiply the exponents in ${(3^{2})}^{2 k - 1}$ .

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Step 4.1

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

$3^{5 k + 10} \cdot 3^{2 (2 k - 1)} = 9$

Step 4.2

Apply the distributive property.

$3^{5 k + 10} \cdot 3^{2 (2 k) + 2 \cdot - 1} = 9$

Step 4.3

Multiply $2$ by $2$ .

$3^{5 k + 10} \cdot 3^{4 k + 2 \cdot - 1} = 9$

Step 4.4

Multiply $2$ by $- 1$ .

$3^{5 k + 10} \cdot 3^{4 k - 2} = 9$

Step 5

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

$3^{5 k + 10 + 4 k - 2} = 9$

Step 6

Add $5 k$ and $4 k$ .

$3^{9 k + 10 - 2} = 9$

Step 7

Subtract $2$ from $10$ .

$3^{9 k + 8} = 9$

Step 8

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

$3^{9 k + 8} = 3^{2}$

Step 9

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

$9 k + 8 = 2$

Step 10

Solve for $k$ .

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Step 10.1

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

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Step 10.1.1

Subtract $8$ from both sides of the equation.

$9 k = 2 - 8$

Step 10.1.2

Subtract $8$ from $2$ .

$9 k = - 6$

Step 10.2

Divide each term in $9 k = - 6$ by $9$ and simplify.

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Step 10.2.1

Divide each term in $9 k = - 6$ by $9$ .

$\frac{9 k}{9} = \frac{- 6}{9}$

Step 10.2.2

Simplify the left side.

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Step 10.2.2.1

Cancel the common factor of $9$ .

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Step 10.2.2.1.1

Cancel the common factor.

$\frac{9 k}{9} = \frac{- 6}{9}$

Step 10.2.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 6}{9}$

Step 10.2.3

Simplify the right side.

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Step 10.2.3.1

Cancel the common factor of $- 6$ and $9$ .

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Step 10.2.3.1.1

Factor $3$ out of $- 6$ .

$k = \frac{3 (- 2)}{9}$

Step 10.2.3.1.2

Cancel the common factors.

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Step 10.2.3.1.2.1

Factor $3$ out of $9$ .

$k = \frac{3 \cdot - 2}{3 \cdot 3}$

Step 10.2.3.1.2.2

Cancel the common factor.

$k = \frac{3 \cdot - 2}{3 \cdot 3}$

Step 10.2.3.1.2.3

Rewrite the expression.

$k = \frac{- 2}{3}$

Step 10.2.3.2

Move the negative in front of the fraction.

$k = - \frac{2}{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$k = - \frac{2}{3}$

Decimal Form:

$k = - 0.\overline{6}$