Solve for d (1/2)^2*4^(3d)=1

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

Step 1

Simplify.

Rewrite $\frac{1}{2}$ as $2^{- 1}$ .

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

Raise $2$ to the power of $1$ .

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

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

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

Multiply $- 1$ by $1$ .

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

Multiply the exponents in ${(2^{- 1})}^{2}$ .

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

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

Multiply $- 1$ by $2$ .

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

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

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

Multiply the exponents in ${(2^{2})}^{3 d}$ .

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

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

Multiply $3$ by $2$ .

$2^{- 2} \cdot 2^{6 d} = 1$

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

$2^{- 2 + 6 d} = 1$

Step 2

Divide each term in $2^{- 2 + 6 d} = 1$ by ${(\frac{1}{2})}^{2}$ and simplify.

Divide each term in $2^{- 2 + 6 d} = 1$ by ${(\frac{1}{2})}^{2}$ .

$\frac{2^{- 2 + 6 d}}{{(\frac{1}{2})}^{2}} = \frac{1}{{(\frac{1}{2})}^{2}}$

Simplify the left side.

Simplify the denominator.

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

$\frac{2^{- 2 + 6 d}}{\frac{1^{2}}{2^{2}}} = \frac{1}{{(\frac{1}{2})}^{2}}$

One to any power is one.

$\frac{2^{- 2 + 6 d}}{\frac{1}{2^{2}}} = \frac{1}{{(\frac{1}{2})}^{2}}$

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

$\frac{2^{- 2 + 6 d}}{\frac{1}{4}} = \frac{1}{{(\frac{1}{2})}^{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2^{- 2 + 6 d} \cdot 4 = \frac{1}{{(\frac{1}{2})}^{2}}$

Multiply $2^{- 2 + 6 d} \cdot 4$ .

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

$2^{- 2 + 6 d} \cdot 2^{2} = \frac{1}{{(\frac{1}{2})}^{2}}$

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

$2^{- 2 + 6 d + 2} = \frac{1}{{(\frac{1}{2})}^{2}}$

Add $- 2$ and $2$ .

$2^{6 d + 0} = \frac{1}{{(\frac{1}{2})}^{2}}$

Add $6 d$ and $0$ .

$2^{6 d} = \frac{1}{{(\frac{1}{2})}^{2}}$

Simplify the right side.

Simplify the denominator.

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

$2^{6 d} = \frac{1}{\frac{1^{2}}{2^{2}}}$

One to any power is one.

$2^{6 d} = \frac{1}{\frac{1}{2^{2}}}$

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

$2^{6 d} = \frac{1}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2^{6 d} = 1 \cdot 4$

Multiply $4$ by $1$ .

$2^{6 d} = 4$

Step 3

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

$2^{6 d} = 2^{2}$

Step 4

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

$6 d = 2$

Step 5

Divide each term in $6 d = 2$ by $6$ and simplify.

Divide each term in $6 d = 2$ by $6$ .

$\frac{6 d}{6} = \frac{2}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 d}{6} = \frac{2}{6}$

Divide $d$ by $1$ .

$d = \frac{2}{6}$

Simplify the right side.

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2$ .

$d = \frac{2 (1)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$d = 0.\overline{3}$

Solve for d (1/2)^2*4^(3d)=1

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