Algebra Examples

$\frac{2}{5} \cdot 4^{5 v} - 8 = 4$

Step 1

Simplify $\frac{2}{5} \cdot 4^{5 v} - 8$ .

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Step 1.1

Multiply $\frac{2}{5} \cdot 4^{5 v}$ .

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Step 1.1.1

Combine $\frac{2}{5}$ and $4^{5 v}$ .

$\frac{2 \cdot 4^{5 v}}{5} - 8 = 4$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{2 \cdot {(2^{2})}^{5 v}}{5} - 8 = 4$

Step 1.1.3

Multiply the exponents in ${(2^{2})}^{5 v}$ .

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Step 1.1.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \cdot 2^{2 (5 v)}}{5} - 8 = 4$

Step 1.1.3.2

Multiply $5$ by $2$ .

$\frac{2 \cdot 2^{10 v}}{5} - 8 = 4$

Step 1.1.4

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

$\frac{2^{1 + 10 v}}{5} - 8 = 4$

Step 1.2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2^{1 + 10 v}}{5} - 8 \cdot \frac{5}{5} = 4$

Step 1.3

Combine $- 8$ and $\frac{5}{5}$ .

$\frac{2^{1 + 10 v}}{5} + \frac{- 8 \cdot 5}{5} = 4$

Step 1.4

Combine the numerators over the common denominator.

$\frac{2^{1 + 10 v} - 8 \cdot 5}{5} = 4$

Step 1.5

Multiply $- 8$ by $5$ .

$\frac{2^{1 + 10 v} - 40}{5} = 4$

Step 2

Multiply both sides by $5$ .

$\frac{2^{1 + 10 v} - 40}{5} \cdot 5 = 4 \cdot 5$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{2^{1 + 10 v} - 40}{5} \cdot 5$ .

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Step 3.1.1.1

Cancel the common factor of $5$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{2^{1 + 10 v} - 40}{5} \cdot 5 = 4 \cdot 5$

Step 3.1.1.1.2

Rewrite the expression.

$2^{1 + 10 v} - 40 = 4 \cdot 5$

Step 3.1.1.2

Reorder $1$ and $10 v$ .

$2^{10 v + 1} - 40 = 4 \cdot 5$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $4$ by $5$ .

$2^{10 v + 1} - 40 = 20$

Step 4

Solve for $v$ .

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Step 4.1

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

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Step 4.1.1

Add $40$ to both sides of the equation.

$2^{10 v + 1} = 20 + 40$

Step 4.1.2

Add $20$ and $40$ .

$2^{10 v + 1} = 60$

Step 4.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (2^{10 v + 1}) = \ln (60)$

Step 4.3

Expand $\ln (2^{10 v + 1})$ by moving $10 v + 1$ outside the logarithm.

$(10 v + 1) \ln (2) = \ln (60)$

Step 4.4

Simplify the left side.

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Step 4.4.1

Simplify $(10 v + 1) \ln (2)$ .

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Step 4.4.1.1

Apply the distributive property.

$10 v \ln (2) + 1 \ln (2) = \ln (60)$

Step 4.4.1.2

Multiply $\ln (2)$ by $1$ .

$10 v \ln (2) + \ln (2) = \ln (60)$

Step 4.5

Move all the terms containing a logarithm to the left side of the equation.

$10 v \ln (2) + \ln (2) - \ln (60) = 0$

Step 4.6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$10 v \ln (2) + \ln (\frac{2}{60}) = 0$

Step 4.7

Cancel the common factor of $2$ and $60$ .

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Step 4.7.1

Factor $2$ out of $2$ .

$10 v \ln (2) + \ln (\frac{2 (1)}{60}) = 0$

Step 4.7.2

Cancel the common factors.

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Step 4.7.2.1

Factor $2$ out of $60$ .

$10 v \ln (2) + \ln (\frac{2 \cdot 1}{2 \cdot 30}) = 0$

Step 4.7.2.2

Cancel the common factor.

$10 v \ln (2) + \ln (\frac{2 \cdot 1}{2 \cdot 30}) = 0$

Step 4.7.2.3

Rewrite the expression.

$10 v \ln (2) + \ln (\frac{1}{30}) = 0$

Step 4.8

Subtract $\ln (\frac{1}{30})$ from both sides of the equation.

$10 v \ln (2) = - \ln (\frac{1}{30})$

Step 4.9

Divide each term in $10 v \ln (2) = - \ln (\frac{1}{30})$ by $10 \ln (2)$ and simplify.

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Step 4.9.1

Divide each term in $10 v \ln (2) = - \ln (\frac{1}{30})$ by $10 \ln (2)$ .

$\frac{10 v \ln (2)}{10 \ln (2)} = \frac{- \ln (\frac{1}{30})}{10 \ln (2)}$

Step 4.9.2

Simplify the left side.

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Step 4.9.2.1

Cancel the common factor of $10$ .

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Step 4.9.2.1.1

Cancel the common factor.

$\frac{10 v \ln (2)}{10 \ln (2)} = \frac{- \ln (\frac{1}{30})}{10 \ln (2)}$

Step 4.9.2.1.2

Rewrite the expression.

$\frac{v \ln (2)}{\ln (2)} = \frac{- \ln (\frac{1}{30})}{10 \ln (2)}$

Step 4.9.2.2

Cancel the common factor of $\ln (2)$ .

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Step 4.9.2.2.1

Cancel the common factor.

$\frac{v \ln (2)}{\ln (2)} = \frac{- \ln (\frac{1}{30})}{10 \ln (2)}$

Step 4.9.2.2.2

Divide $v$ by $1$ .

$v = \frac{- \ln (\frac{1}{30})}{10 \ln (2)}$

Step 4.9.3

Simplify the right side.

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Step 4.9.3.1

Move the negative in front of the fraction.

$v = - \frac{\ln (\frac{1}{30})}{10 \ln (2)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{\ln (\frac{1}{30})}{10 \ln (2)}$

Decimal Form:

$v = 0.49068905 \dots$