Algebra Examples

$4^{x} = 16 \sqrt{128}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{128}$ as $128^{\frac{1}{2}}$ .

$4^{x} = 16 \cdot 128^{\frac{1}{2}}$

Step 2

Rewrite $16$ as $2^{4}$ .

$4^{x} = 2^{4} \cdot 128^{\frac{1}{2}}$

Step 3

Rewrite $128$ as $2^{7}$ .

$4^{x} = 2^{4} \cdot {(2^{7})}^{\frac{1}{2}}$

Step 4

Multiply the exponents in ${(2^{7})}^{\frac{1}{2}}$ .

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Step 4.1

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

$4^{x} = 2^{4} \cdot 2^{7 (\frac{1}{2})}$

Step 4.2

Combine $7$ and $\frac{1}{2}$ .

$4^{x} = 2^{4} \cdot 2^{\frac{7}{2}}$

Step 5

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

$4^{x} = 2^{4 + \frac{7}{2}}$

Step 6

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4^{x} = 2^{4 \cdot \frac{2}{2} + \frac{7}{2}}$

Step 7

Combine $4$ and $\frac{2}{2}$ .

$4^{x} = 2^{\frac{4 \cdot 2}{2} + \frac{7}{2}}$

Step 8

Combine the numerators over the common denominator.

$4^{x} = 2^{\frac{4 \cdot 2 + 7}{2}}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $4$ by $2$ .

$4^{x} = 2^{\frac{8 + 7}{2}}$

Step 9.2

Add $8$ and $7$ .

$4^{x} = 2^{\frac{15}{2}}$

Step 10

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

$2^{2 x} = 2^{\frac{15}{2}}$

Step 11

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

$2 x = \frac{15}{2}$

Step 12

Divide each term in $2 x = \frac{15}{2}$ by $2$ and simplify.

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Step 12.1

Divide each term in $2 x = \frac{15}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{15}{2}}{2}$

Step 12.2

Simplify the left side.

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Step 12.2.1

Cancel the common factor of $2$ .

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Step 12.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{15}{2}}{2}$

Step 12.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{15}{2}}{2}$

Step 12.3

Simplify the right side.

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Step 12.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{15}{2} \cdot \frac{1}{2}$

Step 12.3.2

Multiply $\frac{15}{2} \cdot \frac{1}{2}$ .

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Step 12.3.2.1

Multiply $\frac{15}{2}$ by $\frac{1}{2}$ .

$x = \frac{15}{2 \cdot 2}$

Step 12.3.2.2

Multiply $2$ by $2$ .

$x = \frac{15}{4}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15}{4}$

Decimal Form:

$x = 3.75$

Mixed Number Form:

$x = 3 \frac{3}{4}$