Solve for x 32^(x/5)=(1/16)^(4x-3)

$32^{\frac{x}{5}} = {(\frac{1}{16})}^{4 x - 3}$

Apply the product rule to $\frac{1}{16}$ .

$32^{\frac{x}{5}} = \frac{1^{4 x - 3}}{16^{4 x - 3}}$

One to any power is one.

$32^{\frac{x}{5}} = \frac{1}{16^{4 x - 3}}$

Move $16^{4 x - 3}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$32^{\frac{x}{5}} = 16^{- (4 x - 3)}$

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

$2^{5 \frac{x}{5}} = 2^{4 (- (4 x - 3))}$

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

$5 \frac{x}{5} = 4 (- (4 x - 3))$

Solve for $x$ .

Simplify $5 (\frac{x}{5})$ .

Start simplifying.

$0 + 0 + 5 (\frac{x}{5}) = 4 (- (4 x - 3))$

Simplify by adding zeros.

$5 (\frac{x}{5}) = 4 (- (4 x - 3))$

Cancel the common factor of $5$ .

Cancel the common factor.

$5 \frac{x}{5} = 4 (- (4 x - 3))$

Rewrite the expression.

$x = 4 (- (4 x - 3))$

Simplify $4 (- (4 x - 3))$ .

Apply the distributive property.

$x = 4 (- (4 x) - - 3)$

Multiply.

Multiply $4$ by $- 1$ .

$x = 4 (- 4 x - - 3)$

Multiply $- 1$ by $- 3$ .

$x = 4 (- 4 x + 3)$

Apply the distributive property.

$x = 4 (- 4 x) + 4 \cdot 3$

Multiply.

Multiply $- 4$ by $4$ .

$x = - 16 x + 4 \cdot 3$

Multiply $4$ by $3$ .

$x = - 16 x + 12$

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

Add $16 x$ to both sides of the equation.

$x + 16 x = 12$

Add $x$ and $16 x$ .

$17 x = 12$

Divide each term in $17 x = 12$ by $17$ and simplify.

Divide each term in $17 x = 12$ by $17$ .

$\frac{17 x}{17} = \frac{12}{17}$

Simplify the left side.

Cancel the common factor of $17$ .

Cancel the common factor.

$\frac{17 x}{17} = \frac{12}{17}$

Divide $x$ by $1$ .

$x = \frac{12}{17}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{12}{17}$

Decimal Form:

$x = 0.70588235 \dots$

Solve for x 32^(x/5)=(1/16)^(4x-3)

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