Solve for v 343^(2v+3)=(1/7)^(5-2v)

$343^{2 v + 3} = {(\frac{1}{7})}^{5 - 2 v}$

Step 1

Apply the product rule to $\frac{1}{7}$ .

$343^{2 v + 3} = \frac{1^{5 - 2 v}}{7^{5 - 2 v}}$

Step 2

One to any power is one.

$343^{2 v + 3} = \frac{1}{7^{5 - 2 v}}$

Step 3

Move $7^{5 - 2 v}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$343^{2 v + 3} = 7^{- (5 - 2 v)}$

Step 4

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

$7^{3 (2 v + 3)} = 7^{- (5 - 2 v)}$

Step 5

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

$3 (2 v + 3) = - (5 - 2 v)$

Step 6

Solve for $v$ .

Simplify $3 (2 v + 3)$ .

Rewrite.

$0 + 0 + 3 (2 v + 3) = - (5 - 2 v)$

Simplify by adding zeros.

$3 (2 v + 3) = - (5 - 2 v)$

Apply the distributive property.

$3 (2 v) + 3 \cdot 3 = - (5 - 2 v)$

Multiply.

Multiply $2$ by $3$ .

$6 v + 3 \cdot 3 = - (5 - 2 v)$

Multiply $3$ by $3$ .

$6 v + 9 = - (5 - 2 v)$

Simplify $- (5 - 2 v)$ .

Apply the distributive property.

$6 v + 9 = - 1 \cdot 5 - (- 2 v)$

Multiply.

Multiply $- 1$ by $5$ .

$6 v + 9 = - 5 - (- 2 v)$

Multiply $- 2$ by $- 1$ .

$6 v + 9 = - 5 + 2 v$

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

Subtract $2 v$ from both sides of the equation.

$6 v + 9 - 2 v = - 5$

Subtract $2 v$ from $6 v$ .

$4 v + 9 = - 5$

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

Subtract $9$ from both sides of the equation.

$4 v = - 5 - 9$

Subtract $9$ from $- 5$ .

$4 v = - 14$

Divide each term in $4 v = - 14$ by $4$ and simplify.

Divide each term in $4 v = - 14$ by $4$ .

$\frac{4 v}{4} = \frac{- 14}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 v}{4} = \frac{- 14}{4}$

Divide $v$ by $1$ .

$v = \frac{- 14}{4}$

Simplify the right side.

Cancel the common factor of $- 14$ and $4$ .

Factor $2$ out of $- 14$ .

$v = \frac{2 (- 7)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$v = \frac{2 \cdot - 7}{2 \cdot 2}$

Cancel the common factor.

$v = \frac{2 \cdot - 7}{2 \cdot 2}$

Rewrite the expression.

$v = \frac{- 7}{2}$

Move the negative in front of the fraction.

$v = - \frac{7}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{7}{2}$

Decimal Form:

$v = - 3.5$

Mixed Number Form:

$v = - 3 \frac{1}{2}$

Solve for v 343^(2v+3)=(1/7)^(5-2v)

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