Precalculus Examples

$64^{- 3 k - 1} \cdot 4^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 1

Rewrite $64$ as $2^{6}$ .

${(2^{6})}^{- 3 k - 1} \cdot 4^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 2

Multiply the exponents in ${(2^{6})}^{- 3 k - 1}$ .

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Step 2.1

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

$2^{6 (- 3 k - 1)} \cdot 4^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 2.2

Apply the distributive property.

$2^{6 (- 3 k) + 6 \cdot - 1} \cdot 4^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 2.3

Multiply $- 3$ by $6$ .

$2^{- 18 k + 6 \cdot - 1} \cdot 4^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 2.4

Multiply $6$ by $- 1$ .

$2^{- 18 k - 6} \cdot 4^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 3

Rewrite $4$ as $2^{2}$ .

$2^{- 18 k - 6} \cdot {(2^{2})}^{- 2 k + 3} = 16^{- 3 k - 2}$

Step 4

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

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Step 4.1

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

$2^{- 18 k - 6} \cdot 2^{2 (- 2 k + 3)} = 16^{- 3 k - 2}$

Step 4.2

Apply the distributive property.

$2^{- 18 k - 6} \cdot 2^{2 (- 2 k) + 2 \cdot 3} = 16^{- 3 k - 2}$

Step 4.3

Multiply $- 2$ by $2$ .

$2^{- 18 k - 6} \cdot 2^{- 4 k + 2 \cdot 3} = 16^{- 3 k - 2}$

Step 4.4

Multiply $2$ by $3$ .

$2^{- 18 k - 6} \cdot 2^{- 4 k + 6} = 16^{- 3 k - 2}$

Step 5

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

$2^{- 18 k - 6 - 4 k + 6} = 16^{- 3 k - 2}$

Step 6

Subtract $4 k$ from $- 18 k$ .

$2^{- 22 k - 6 + 6} = 16^{- 3 k - 2}$

Step 7

Add $- 6$ and $6$ .

$2^{- 22 k + 0} = 16^{- 3 k - 2}$

Step 8

Add $- 22 k$ and $0$ .

$2^{- 22 k} = 16^{- 3 k - 2}$

Step 9

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

$2^{- 22 k} = 2^{4 (- 3 k - 2)}$

Step 10

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

$- 22 k = 4 (- 3 k - 2)$

Step 11

Solve for $k$ .

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Step 11.1

Simplify $4 (- 3 k - 2)$ .

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Step 11.1.1

Apply the distributive property.

$- 22 k = 4 (- 3 k) + 4 \cdot - 2$

Step 11.1.2

Multiply.

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Step 11.1.2.1

Multiply $- 3$ by $4$ .

$- 22 k = - 12 k + 4 \cdot - 2$

Step 11.1.2.2

Multiply $4$ by $- 2$ .

$- 22 k = - 12 k - 8$

Step 11.2

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

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Step 11.2.1

Add $12 k$ to both sides of the equation.

$- 22 k + 12 k = - 8$

Step 11.2.2

Add $- 22 k$ and $12 k$ .

$- 10 k = - 8$

Step 11.3

Divide each term in $- 10 k = - 8$ by $- 10$ and simplify.

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Step 11.3.1

Divide each term in $- 10 k = - 8$ by $- 10$ .

$\frac{- 10 k}{- 10} = \frac{- 8}{- 10}$

Step 11.3.2

Simplify the left side.

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Step 11.3.2.1

Cancel the common factor of $- 10$ .

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Step 11.3.2.1.1

Cancel the common factor.

$\frac{- 10 k}{- 10} = \frac{- 8}{- 10}$

Step 11.3.2.1.2

Divide $k$ by $1$ .

$k = \frac{- 8}{- 10}$

Step 11.3.3

Simplify the right side.

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Step 11.3.3.1

Cancel the common factor of $- 8$ and $- 10$ .

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Step 11.3.3.1.1

Factor $- 2$ out of $- 8$ .

$k = \frac{- 2 (4)}{- 10}$

Step 11.3.3.1.2

Cancel the common factors.

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Step 11.3.3.1.2.1

Factor $- 2$ out of $- 10$ .

$k = \frac{- 2 \cdot 4}{- 2 \cdot 5}$

Step 11.3.3.1.2.2

Cancel the common factor.

$k = \frac{- 2 \cdot 4}{- 2 \cdot 5}$

Step 11.3.3.1.2.3

Rewrite the expression.

$k = \frac{4}{5}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$k = \frac{4}{5}$

Decimal Form:

$k = 0.8$