Pre-Algebra Examples

$0.125 \cdot 4^{2 x - 8} = {(0.25 \div \sqrt{2})}^{- x}$

Step 1

Rewrite the division as a fraction.

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25}{\sqrt{2}})}^{- x}$

Step 2

Multiply $\frac{0.25}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{- x}$

Step 3

Combine and simplify the denominator.

Tap for more steps...

Step 3.1

Multiply $\frac{0.25}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{- x}$

Step 3.2

Raise $\sqrt{2}$ to the power of $1$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{- x}$

Step 3.3

Raise $\sqrt{2}$ to the power of $1$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{- x}$

Step 3.4

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

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{- x}$

Step 3.5

Add $1$ and $1$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{{\sqrt{2}}^{2}})}^{- x}$

Step 3.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Tap for more steps...

Step 3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{- x}$

Step 3.6.2

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

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{- x}$

Step 3.6.3

Combine $\frac{1}{2}$ and $2$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{2^{\frac{2}{2}}})}^{- x}$

Step 3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.6.4.1

Cancel the common factor.

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{2^{\frac{2}{2}}})}^{- x}$

Step 3.6.4.2

Rewrite the expression.

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{2^{1}})}^{- x}$

Step 3.6.5

Evaluate the exponent.

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.25 \sqrt{2}}{2})}^{- x}$

Step 4

Multiply $0.25$ by $\sqrt{2}$ .

$0.125 \cdot 4^{2 x - 8} = {(\frac{0.35355339}{2})}^{- x}$

Step 5

Divide $0.35355339$ by $2$ .

$0.125 \cdot 4^{2 x - 8} = {0.17677669}^{- x}$

Step 6

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

$\ln (0.125 \cdot 4^{2 x - 8}) = \ln ({0.17677669}^{- x})$

Step 7

Expand the left side.

Tap for more steps...

Step 7.1

Rewrite $\ln (0.125 \cdot 4^{2 x - 8})$ as $\ln (0.125) + \ln (4^{2 x - 8})$ .

$\ln (0.125) + \ln (4^{2 x - 8}) = \ln ({0.17677669}^{- x})$

Step 7.2

Expand $\ln (4^{2 x - 8})$ by moving $2 x - 8$ outside the logarithm.

$\ln (0.125) + (2 x - 8) \ln (4) = \ln ({0.17677669}^{- x})$

Step 8

Expand $\ln ({0.17677669}^{- x})$ by moving $- x$ outside the logarithm.

$\ln (0.125) + (2 x - 8) \ln (4) = - x \ln (0.17677669)$

Step 9

Simplify the left side.

Tap for more steps...

Step 9.1

Apply the distributive property.

$\ln (0.125) + 2 x \ln (4) - 8 \ln (4) = - x \ln (0.17677669)$

Step 10

Reorder $\ln (0.125)$ and $2 x \ln (4)$ .

$2 x \ln (4) + \ln (0.125) - 8 \ln (4) = - x \ln (0.17677669)$

Step 11

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

$2 x \ln (4) + \ln (0.125) - 8 \ln (4) + x \ln (0.17677669) = 0$

Step 12

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

Tap for more steps...

Step 12.1

Subtract $\ln (0.125)$ from both sides of the equation.

$2 x \ln (4) - 8 \ln (4) + x \ln (0.17677669) = - \ln (0.125)$

Step 12.2

Add $8 \ln (4)$ to both sides of the equation.

$2 x \ln (4) + x \ln (0.17677669) = - \ln (0.125) + 8 \ln (4)$

Step 13

Factor $x$ out of $2 x \ln (4) + x \ln (0.17677669)$ .

Tap for more steps...

Step 13.1

Factor $x$ out of $2 x \ln (4)$ .

$x (2 \ln (4)) + x \ln (0.17677669) = - \ln (0.125) + 8 \ln (4)$

Step 13.2

Factor $x$ out of $x \ln (0.17677669)$ .

$x (2 \ln (4)) + x (\ln (0.17677669)) = - \ln (0.125) + 8 \ln (4)$

Step 13.3

Factor $x$ out of $x (2 \ln (4)) + x (\ln (0.17677669))$ .

$x (2 \ln (4) + \ln (0.17677669)) = - \ln (0.125) + 8 \ln (4)$

Step 14

Divide each term in $x (2 \ln (4) + \ln (0.17677669)) = - \ln (0.125) + 8 \ln (4)$ by $2 \ln (4) + \ln (0.17677669)$ and simplify.

Tap for more steps...

Step 14.1

Divide each term in $x (2 \ln (4) + \ln (0.17677669)) = - \ln (0.125) + 8 \ln (4)$ by $2 \ln (4) + \ln (0.17677669)$ .

$\frac{x (2 \ln (4) + \ln (0.17677669))}{2 \ln (4) + \ln (0.17677669)} = \frac{- \ln (0.125)}{2 \ln (4) + \ln (0.17677669)} + \frac{8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Step 14.2

Simplify the left side.

Tap for more steps...

Step 14.2.1

Cancel the common factor of $2 \ln (4) + \ln (0.17677669)$ .

Tap for more steps...

Step 14.2.1.1

Cancel the common factor.

$\frac{x (2 \ln (4) + \ln (0.17677669))}{2 \ln (4) + \ln (0.17677669)} = \frac{- \ln (0.125)}{2 \ln (4) + \ln (0.17677669)} + \frac{8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Step 14.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \ln (0.125)}{2 \ln (4) + \ln (0.17677669)} + \frac{8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Step 14.3

Simplify the right side.

Tap for more steps...

Step 14.3.1

Combine the numerators over the common denominator.

$x = \frac{- \ln (0.125) + 8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Step 14.3.2

Factor $- 1$ out of $- \ln (0.125)$ .

$x = \frac{- \ln (0.125) + 8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Step 14.3.3

Factor $- 1$ out of $8 \ln (4)$ .

$x = \frac{- \ln (0.125) - (- 8 \ln (4))}{2 \ln (4) + \ln (0.17677669)}$

Step 14.3.4

Factor $- 1$ out of $- \ln (0.125) - (- 8 \ln (4))$ .

$x = \frac{- (\ln (0.125) - 8 \ln (4))}{2 \ln (4) + \ln (0.17677669)}$

Step 14.3.5

Simplify the expression.

Tap for more steps...

Step 14.3.5.1

Rewrite $- (\ln (0.125) - 8 \ln (4))$ as $- 1 (\ln (0.125) - 8 \ln (4))$ .

$x = \frac{- 1 (\ln (0.125) - 8 \ln (4))}{2 \ln (4) + \ln (0.17677669)}$

Step 14.3.5.2

Move the negative in front of the fraction.

$x = - \frac{\ln (0.125) - 8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (0.125) - 8 \ln (4)}{2 \ln (4) + \ln (0.17677669)}$

Decimal Form:

$x = 12.\overline{6}$