Precalculus Examples

${(\frac{1}{4})}^{3 z - 1} = 16^{z + 2} \cdot 64^{z - 2}$

Step 1

Apply the product rule to $\frac{1}{4}$ .

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

Step 2

One to any power is one.

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

Step 3

Move $4^{3 z - 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$4^{- (3 z - 1)} = 16^{z + 2} \cdot 64^{z - 2}$

Step 4

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

$4^{- (3 z - 1)} = {(2^{4})}^{z + 2} \cdot 64^{z - 2}$

Step 5

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

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Step 5.1

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

$4^{- (3 z - 1)} = 2^{4 (z + 2)} \cdot 64^{z - 2}$

Step 5.2

Apply the distributive property.

$4^{- (3 z - 1)} = 2^{4 z + 4 \cdot 2} \cdot 64^{z - 2}$

Step 5.3

Multiply $4$ by $2$ .

$4^{- (3 z - 1)} = 2^{4 z + 8} \cdot 64^{z - 2}$

Step 6

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

$4^{- (3 z - 1)} = 2^{4 z + 8} \cdot {(2^{6})}^{z - 2}$

Step 7

Multiply the exponents in ${(2^{6})}^{z - 2}$ .

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Step 7.1

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

$4^{- (3 z - 1)} = 2^{4 z + 8} \cdot 2^{6 (z - 2)}$

Step 7.2

Apply the distributive property.

$4^{- (3 z - 1)} = 2^{4 z + 8} \cdot 2^{6 z + 6 \cdot - 2}$

Step 7.3

Multiply $6$ by $- 2$ .

$4^{- (3 z - 1)} = 2^{4 z + 8} \cdot 2^{6 z - 12}$

Step 8

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

$4^{- (3 z - 1)} = 2^{4 z + 8 + 6 z - 12}$

Step 9

Add $4 z$ and $6 z$ .

$4^{- (3 z - 1)} = 2^{10 z + 8 - 12}$

Step 10

Subtract $12$ from $8$ .

$4^{- (3 z - 1)} = 2^{10 z - 4}$

Step 11

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

$2^{2 (- (3 z - 1))} = 2^{10 z - 4}$

Step 12

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

$2 (- (3 z - 1)) = 10 z - 4$

Step 13

Solve for $z$ .

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Step 13.1

Simplify $2 (- (3 z - 1))$ .

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Step 13.1.1

Rewrite.

$0 + 0 + 2 (- (3 z - 1)) = 10 z - 4$

Step 13.1.2

Simplify by adding zeros.

$2 (- (3 z - 1)) = 10 z - 4$

Step 13.1.3

Apply the distributive property.

$2 (- (3 z) - - 1) = 10 z - 4$

Step 13.1.4

Multiply.

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Step 13.1.4.1

Multiply $3$ by $- 1$ .

$2 (- 3 z - - 1) = 10 z - 4$

Step 13.1.4.2

Multiply $- 1$ by $- 1$ .

$2 (- 3 z + 1) = 10 z - 4$

Step 13.1.5

Apply the distributive property.

$2 (- 3 z) + 2 \cdot 1 = 10 z - 4$

Step 13.1.6

Multiply.

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Step 13.1.6.1

Multiply $- 3$ by $2$ .

$- 6 z + 2 \cdot 1 = 10 z - 4$

Step 13.1.6.2

Multiply $2$ by $1$ .

$- 6 z + 2 = 10 z - 4$

Step 13.2

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

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Step 13.2.1

Subtract $10 z$ from both sides of the equation.

$- 6 z + 2 - 10 z = - 4$

Step 13.2.2

Subtract $10 z$ from $- 6 z$ .

$- 16 z + 2 = - 4$

Step 13.3

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

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Step 13.3.1

Subtract $2$ from both sides of the equation.

$- 16 z = - 4 - 2$

Step 13.3.2

Subtract $2$ from $- 4$ .

$- 16 z = - 6$

Step 13.4

Divide each term in $- 16 z = - 6$ by $- 16$ and simplify.

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Step 13.4.1

Divide each term in $- 16 z = - 6$ by $- 16$ .

$\frac{- 16 z}{- 16} = \frac{- 6}{- 16}$

Step 13.4.2

Simplify the left side.

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Step 13.4.2.1

Cancel the common factor of $- 16$ .

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Step 13.4.2.1.1

Cancel the common factor.

$\frac{- 16 z}{- 16} = \frac{- 6}{- 16}$

Step 13.4.2.1.2

Divide $z$ by $1$ .

$z = \frac{- 6}{- 16}$

Step 13.4.3

Simplify the right side.

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Step 13.4.3.1

Cancel the common factor of $- 6$ and $- 16$ .

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Step 13.4.3.1.1

Factor $- 2$ out of $- 6$ .

$z = \frac{- 2 (3)}{- 16}$

Step 13.4.3.1.2

Cancel the common factors.

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Step 13.4.3.1.2.1

Factor $- 2$ out of $- 16$ .

$z = \frac{- 2 \cdot 3}{- 2 \cdot 8}$

Step 13.4.3.1.2.2

Cancel the common factor.

$z = \frac{- 2 \cdot 3}{- 2 \cdot 8}$

Step 13.4.3.1.2.3

Rewrite the expression.

$z = \frac{3}{8}$

Step 14

The result can be shown in multiple forms.

Exact Form:

$z = \frac{3}{8}$

Decimal Form:

$z = 0.375$