Solve for v 10^-410^9=10^(v+4)10^(2v-11)

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Solve for v 10^-4*10^9=10^(v+4)*10^(2v-11)

$10^{- 4} \cdot 10^{9} = 10^{v + 4} \cdot 10^{2 v - 11}$

Rewrite the equation as $10^{v + 4} \cdot 10^{2 v - 11} = 10^{- 4} \cdot 10^{9}$ .

$10^{v + 4} \cdot 10^{2 v - 11} = 10^{- 4} \cdot 10^{9}$

Multiply $10^{v + 4}$ by $10^{2 v - 11}$ by adding the exponents.

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

$10^{v + 4 + 2 v - 11} = 10^{- 4} \cdot 10^{9}$

Add $v$ and $2 v$ .

$10^{3 v + 4 - 11} = 10^{- 4} \cdot 10^{9}$

Subtract $11$ from $4$ .

$10^{3 v - 7} = 10^{- 4} \cdot 10^{9}$

Multiply $10^{- 4}$ by $10^{9}$ by adding the exponents.

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

$10^{3 v - 7} = 10^{- 4 + 9}$

Add $- 4$ and $9$ .

$10^{3 v - 7} = 10^{5}$

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

$3 v - 7 = 5$

Solve for $v$ .

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

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Add $7$ to both sides of the equation.

$3 v = 5 + 7$

Add $5$ and $7$ .

$3 v = 12$

Divide each term in $3 v = 12$ by $3$ and simplify.

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Divide each term in $3 v = 12$ by $3$ .

$\frac{3 v}{3} = \frac{12}{3}$

Simplify the left side.

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Cancel the common factor of $3$ .

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Cancel the common factor.

$\frac{3 v}{3} = \frac{12}{3}$

Divide $v$ by $1$ .

$v = \frac{12}{3}$

Simplify the right side.

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Divide $12$ by $3$ .

$v = 4$

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