Algebra Examples

$\frac{1}{64} = 4^{2 s - 1} \cdot 16^{2 s + 2}$

Step 1

Rewrite the equation as $4^{2 s - 1} \cdot 16^{2 s + 2} = \frac{1}{64}$ .

$4^{2 s - 1} \cdot 16^{2 s + 2} = \frac{1}{64}$

Step 2

Raise $64$ to the power of $1$ .

$4^{2 s - 1} \cdot 16^{2 s + 2} = \frac{1}{64^{1}}$

Step 3

Move $64^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 4

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

${(2^{2})}^{2 s - 1} \cdot 16^{2 s + 2} = 64^{- 1}$

Step 5

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

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Step 5.1

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

$2^{2 (2 s - 1)} \cdot 16^{2 s + 2} = 64^{- 1}$

Step 5.2

Apply the distributive property.

$2^{2 (2 s) + 2 \cdot - 1} \cdot 16^{2 s + 2} = 64^{- 1}$

Step 5.3

Multiply $2$ by $2$ .

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

Step 5.4

Multiply $2$ by $- 1$ .

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

Step 6

Rewrite $16$ as $2^{4}$ .

$2^{4 s - 2} \cdot {(2^{4})}^{2 s + 2} = 64^{- 1}$

Step 7

Multiply the exponents in ${(2^{4})}^{2 s + 2}$ .

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Step 7.1

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

$2^{4 s - 2} \cdot 2^{4 (2 s + 2)} = 64^{- 1}$

Step 7.2

Apply the distributive property.

$2^{4 s - 2} \cdot 2^{4 (2 s) + 4 \cdot 2} = 64^{- 1}$

Step 7.3

Multiply $2$ by $4$ .

$2^{4 s - 2} \cdot 2^{8 s + 4 \cdot 2} = 64^{- 1}$

Step 7.4

Multiply $4$ by $2$ .

$2^{4 s - 2} \cdot 2^{8 s + 8} = 64^{- 1}$

Step 8

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

$2^{4 s - 2 + 8 s + 8} = 64^{- 1}$

Step 9

Add $4 s$ and $8 s$ .

$2^{12 s - 2 + 8} = 64^{- 1}$

Step 10

Add $- 2$ and $8$ .

$2^{12 s + 6} = 64^{- 1}$

Step 11

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

$2^{12 s + 6} = 2^{- 6}$

Step 12

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

$12 s + 6 = - 6$

Step 13

Solve for $s$ .

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Step 13.1

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

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Step 13.1.1

Subtract $6$ from both sides of the equation.

$12 s = - 6 - 6$

Step 13.1.2

Subtract $6$ from $- 6$ .

$12 s = - 12$

Step 13.2

Divide each term in $12 s = - 12$ by $12$ and simplify.

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Step 13.2.1

Divide each term in $12 s = - 12$ by $12$ .

$\frac{12 s}{12} = \frac{- 12}{12}$

Step 13.2.2

Simplify the left side.

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Step 13.2.2.1

Cancel the common factor of $12$ .

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Step 13.2.2.1.1

Cancel the common factor.

$\frac{12 s}{12} = \frac{- 12}{12}$

Step 13.2.2.1.2

Divide $s$ by $1$ .

$s = \frac{- 12}{12}$

Step 13.2.3

Simplify the right side.

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Step 13.2.3.1

Divide $- 12$ by $12$ .

$s = - 1$