Solve for z 16^(2z+1)=8^(3-4z)

$16^{2 z + 1} = 8^{3 - 4 z}$

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

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

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

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

Solve for $z$ .

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Simplify $4 (2 z + 1)$ .

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Apply the distributive property.

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

Multiply.

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Multiply $2$ by $4$ .

$8 z + 4 \cdot 1 = 3 (3 - 4 z)$

Multiply $4$ by $1$ .

$8 z + 4 = 3 (3 - 4 z)$

Simplify $3 (3 - 4 z)$ .

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Apply the distributive property.

$8 z + 4 = 3 \cdot 3 + 3 (- 4 z)$

Multiply.

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Multiply $3$ by $3$ .

$8 z + 4 = 9 + 3 (- 4 z)$

Multiply $- 4$ by $3$ .

$8 z + 4 = 9 - 12 z$

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

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

$8 z + 4 + 12 z = 9$

Add $8 z$ and $12 z$ .

$20 z + 4 = 9$

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

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Subtract $4$ from both sides of the equation.

$20 z = 9 - 4$

Subtract $4$ from $9$ .

$20 z = 5$

Divide each term by $20$ and simplify.

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Divide each term in $20 z = 5$ by $20$ .

$\frac{20 z}{20} = \frac{5}{20}$

Cancel the common factor of $20$ .

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Cancel the common factor.

$\frac{20 z}{20} = \frac{5}{20}$

Divide $z$ by $1$ .

$z = \frac{5}{20}$

Cancel the common factor of $5$ and $20$ .

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Factor $5$ out of $5$ .

$z = \frac{5 (1)}{20}$

Cancel the common factors.

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Factor $5$ out of $20$ .

$z = \frac{5 \cdot 1}{5 \cdot 4}$

Cancel the common factor.

$z = \frac{5 \cdot 1}{5 \cdot 4}$

Rewrite the expression.

$z = \frac{1}{4}$

The result can be shown in multiple forms.

Exact Form:

$z = \frac{1}{4}$

Decimal Form:

$z = 0.25$

Solve for z 16^(2z+1)=8^(3-4z)

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