Basic Math Examples

$\frac{2^{y + 1} \cdot 3^{2}}{2^{2}} = 3$

Step 1

Cancel the common factor of $2^{y + 1}$ and $2^{2}$ .

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Step 1.1

Factor $2^{2}$ out of $2^{y + 1} \cdot 3^{2}$ .

$\frac{2^{2} (2^{y - 1} \cdot 3^{2})}{2^{2}} = 3$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Multiply by $1$ .

$\frac{2^{2} (2^{y - 1} \cdot 3^{2})}{2^{2} \cdot 1} = 3$

Step 1.2.2

Cancel the common factor.

$\frac{2^{2} (2^{y - 1} \cdot 3^{2})}{2^{2} \cdot 1} = 3$

Step 1.2.3

Rewrite the expression.

$\frac{2^{y - 1} \cdot 3^{2}}{1} = 3$

Step 1.2.4

Divide $2^{y - 1} \cdot 3^{2}$ by $1$ .

$2^{y - 1} \cdot 3^{2} = 3$

Step 2

Solve for $y$ .

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Step 2.1

Divide each term in $2^{y - 1} \cdot 3^{2} = 3$ by $3^{2}$ and simplify.

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Step 2.1.1

Divide each term in $2^{y - 1} \cdot 3^{2} = 3$ by $3^{2}$ .

$\frac{2^{y - 1} \cdot 3^{2}}{3^{2}} = \frac{3}{3^{2}}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $3^{2}$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{2^{y - 1} \cdot 3^{2}}{3^{2}} = \frac{3}{3^{2}}$

Step 2.1.2.1.2

Divide $2^{y - 1}$ by $1$ .

$2^{y - 1} = \frac{3}{3^{2}}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Cancel the common factor of $3$ and $3^{2}$ .

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Step 2.1.3.1.1

Factor $3$ out of $3$ .

$2^{y - 1} = \frac{3 (1)}{3^{2}}$

Step 2.1.3.1.2

Cancel the common factors.

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Step 2.1.3.1.2.1

Factor $3$ out of $3^{2}$ .

$2^{y - 1} = \frac{3 (1)}{3 \cdot 3}$

Step 2.1.3.1.2.2

Cancel the common factor.

$2^{y - 1} = \frac{3 \cdot 1}{3 \cdot 3}$

Step 2.1.3.1.2.3

Rewrite the expression.

$2^{y - 1} = \frac{1}{3}$

Step 2.2

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

$\ln (2^{y - 1}) = \ln (\frac{1}{3})$

Step 2.3

Expand $\ln (2^{y - 1})$ by moving $y - 1$ outside the logarithm.

$(y - 1) \ln (2) = \ln (\frac{1}{3})$

Step 2.4

Simplify the left side.

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Step 2.4.1

Simplify $(y - 1) \ln (2)$ .

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Step 2.4.1.1

Apply the distributive property.

$y \ln (2) - 1 \ln (2) = \ln (\frac{1}{3})$

Step 2.4.1.2

Rewrite $- 1 \ln (2)$ as $- \ln (2)$ .

$y \ln (2) - \ln (2) = \ln (\frac{1}{3})$

Step 2.5

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

$y \ln (2) - \ln (2) - \ln (\frac{1}{3}) = 0$

Step 2.6

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

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Step 2.6.1

Add $\ln (2)$ to both sides of the equation.

$y \ln (2) - \ln (\frac{1}{3}) = \ln (2)$

Step 2.6.2

Add $\ln (\frac{1}{3})$ to both sides of the equation.

$y \ln (2) = \ln (2) + \ln (\frac{1}{3})$

Step 2.7

Divide each term in $y \ln (2) = \ln (2) + \ln (\frac{1}{3})$ by $\ln (2)$ and simplify.

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Step 2.7.1

Divide each term in $y \ln (2) = \ln (2) + \ln (\frac{1}{3})$ by $\ln (2)$ .

$\frac{y \ln (2)}{\ln (2)} = \frac{\ln (2)}{\ln (2)} + \frac{\ln (\frac{1}{3})}{\ln (2)}$

Step 2.7.2

Simplify the left side.

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Step 2.7.2.1

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

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Step 2.7.2.1.1

Cancel the common factor.

$\frac{y \ln (2)}{\ln (2)} = \frac{\ln (2)}{\ln (2)} + \frac{\ln (\frac{1}{3})}{\ln (2)}$

Step 2.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (2)}{\ln (2)} + \frac{\ln (\frac{1}{3})}{\ln (2)}$

Step 2.7.3

Simplify the right side.

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Step 2.7.3.1

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

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Step 2.7.3.1.1

Cancel the common factor.

$y = \frac{\ln (2)}{\ln (2)} + \frac{\ln (\frac{1}{3})}{\ln (2)}$

Step 2.7.3.1.2

Rewrite the expression.

$y = 1 + \frac{\ln (\frac{1}{3})}{\ln (2)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$y = 1 + \frac{\ln (\frac{1}{3})}{\ln (2)}$

Decimal Form:

$y = - 0.58496250 \dots$