Basic Math Examples

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

Step 1

Apply the product rule to $\frac{1}{64}$ .

$\frac{1^{3 v - 2}}{64^{3 v - 2}} = 16^{2 v + 1}$

Step 2

One to any power is one.

$\frac{1}{64^{3 v - 2}} = 16^{2 v + 1}$

Step 3

Move $64^{3 v - 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$64^{- (3 v - 2)} = 16^{2 v + 1}$

Step 4

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

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

Step 5

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

$3 (- (3 v - 2)) = 2 (2 v + 1)$

Step 6

Solve for $v$ .

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Step 6.1

Simplify $3 (- (3 v - 2))$ .

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Step 6.1.1

Rewrite.

$0 + 0 + 3 (- (3 v - 2)) = 2 (2 v + 1)$

Step 6.1.2

Simplify by adding zeros.

$3 (- (3 v - 2)) = 2 (2 v + 1)$

Step 6.1.3

Apply the distributive property.

$3 (- (3 v) - - 2) = 2 (2 v + 1)$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $3$ by $- 1$ .

$3 (- 3 v - - 2) = 2 (2 v + 1)$

Step 6.1.4.2

Multiply $- 1$ by $- 2$ .

$3 (- 3 v + 2) = 2 (2 v + 1)$

Step 6.1.5

Apply the distributive property.

$3 (- 3 v) + 3 \cdot 2 = 2 (2 v + 1)$

Step 6.1.6

Multiply.

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Step 6.1.6.1

Multiply $- 3$ by $3$ .

$- 9 v + 3 \cdot 2 = 2 (2 v + 1)$

Step 6.1.6.2

Multiply $3$ by $2$ .

$- 9 v + 6 = 2 (2 v + 1)$

Step 6.2

Simplify $2 (2 v + 1)$ .

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Step 6.2.1

Apply the distributive property.

$- 9 v + 6 = 2 (2 v) + 2 \cdot 1$

Step 6.2.2

Multiply.

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Step 6.2.2.1

Multiply $2$ by $2$ .

$- 9 v + 6 = 4 v + 2 \cdot 1$

Step 6.2.2.2

Multiply $2$ by $1$ .

$- 9 v + 6 = 4 v + 2$

Step 6.3

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

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Step 6.3.1

Subtract $4 v$ from both sides of the equation.

$- 9 v + 6 - 4 v = 2$

Step 6.3.2

Subtract $4 v$ from $- 9 v$ .

$- 13 v + 6 = 2$

Step 6.4

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

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Step 6.4.1

Subtract $6$ from both sides of the equation.

$- 13 v = 2 - 6$

Step 6.4.2

Subtract $6$ from $2$ .

$- 13 v = - 4$

Step 6.5

Divide each term in $- 13 v = - 4$ by $- 13$ and simplify.

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Step 6.5.1

Divide each term in $- 13 v = - 4$ by $- 13$ .

$\frac{- 13 v}{- 13} = \frac{- 4}{- 13}$

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $- 13$ .

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Step 6.5.2.1.1

Cancel the common factor.

$\frac{- 13 v}{- 13} = \frac{- 4}{- 13}$

Step 6.5.2.1.2

Divide $v$ by $1$ .

$v = \frac{- 4}{- 13}$

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Dividing two negative values results in a positive value.

$v = \frac{4}{13}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = \frac{4}{13}$

Decimal Form:

$v = 0.\overline{307692}$