Algebra Examples

${(\frac{1}{2})}^{2} \cdot 4^{3 d} = 1$

Step 1

Simplify ${(\frac{1}{2})}^{2} \cdot 4^{3 d}$ .

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Step 1.1

Simplify the expression.

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Step 1.1.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{2}}{2^{2}} \cdot 4^{3 d} = 1$

Step 1.1.2

One to any power is one.

$\frac{1}{2^{2}} \cdot 4^{3 d} = 1$

Step 1.1.3

Raise $2$ to the power of $2$ .

$\frac{1}{4} \cdot 4^{3 d} = 1$

Step 1.2

Combine $\frac{1}{4}$ and $4^{3 d}$ .

$\frac{4^{3 d}}{4} = 1$

Step 1.3

Cancel the common factor of $4^{3 d}$ and $4$ .

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Step 1.3.1

Factor $4$ out of $4^{3 d}$ .

$\frac{4 \cdot 4^{3 d - 1}}{4} = 1$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 4^{3 d - 1}}{4 (1)} = 1$

Step 1.3.2.2

Cancel the common factor.

$\frac{4 \cdot 4^{3 d - 1}}{4 \cdot 1} = 1$

Step 1.3.2.3

Rewrite the expression.

$\frac{4^{3 d - 1}}{1} = 1$

Step 1.3.2.4

Divide $4^{3 d - 1}$ by $1$ .

$4^{3 d - 1} = 1$

Step 2

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

$\ln (4^{3 d - 1}) = \ln (1)$

Step 3

Expand $\ln (4^{3 d - 1})$ by moving $3 d - 1$ outside the logarithm.

$(3 d - 1) \ln (4) = \ln (1)$

Step 4

Simplify the left side.

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Step 4.1

Simplify $(3 d - 1) \ln (4)$ .

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Step 4.1.1

Apply the distributive property.

$3 d \ln (4) - 1 \ln (4) = \ln (1)$

Step 4.1.2

Rewrite $- 1 \ln (4)$ as $- \ln (4)$ .

$3 d \ln (4) - \ln (4) = \ln (1)$

Step 5

Simplify the right side.

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Step 5.1

The natural logarithm of $1$ is $0$ .

$3 d \ln (4) - \ln (4) = 0$

Step 6

Add $\ln (4)$ to both sides of the equation.

$3 d \ln (4) = \ln (4)$

Step 7

Divide each term in $3 d \ln (4) = \ln (4)$ by $3 \ln (4)$ and simplify.

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Step 7.1

Divide each term in $3 d \ln (4) = \ln (4)$ by $3 \ln (4)$ .

$\frac{3 d \ln (4)}{3 \ln (4)} = \frac{\ln (4)}{3 \ln (4)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{3 d \ln (4)}{3 \ln (4)} = \frac{\ln (4)}{3 \ln (4)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{d \ln (4)}{\ln (4)} = \frac{\ln (4)}{3 \ln (4)}$

Step 7.2.2

Cancel the common factor of $\ln (4)$ .

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Step 7.2.2.1

Cancel the common factor.

$\frac{d \ln (4)}{\ln (4)} = \frac{\ln (4)}{3 \ln (4)}$

Step 7.2.2.2

Divide $d$ by $1$ .

$d = \frac{\ln (4)}{3 \ln (4)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (4)$ .

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Step 7.3.1.1

Cancel the common factor.

$d = \frac{\ln (4)}{3 \ln (4)}$

Step 7.3.1.2

Rewrite the expression.

$d = \frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$d = \frac{1}{3}$

Decimal Form:

$d = 0.\overline{3}$