Finite Math Examples

$8^{x} \cdot \sqrt{2} = 4^{x}$

Step 1

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

$8^{x} \cdot 2^{\frac{1}{2}} = 4^{x}$

Step 2

Rewrite $8$ as $2^{3}$ .

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

Step 3

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

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

Step 4

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

$2^{3 x + \frac{1}{2}} = 4^{x}$

Step 5

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

$2^{3 x + \frac{1}{2}} = 2^{2 x}$

Step 6

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

$3 x + \frac{1}{2} = 2 x$

Step 7

Solve for $x$ .

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Step 7.1

Move all terms containing $x$ to the left side of the equation.

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Step 7.1.1

Subtract $2 x$ from both sides of the equation.

$3 x + \frac{1}{2} - 2 x = 0$

Step 7.1.2

Subtract $2 x$ from $3 x$ .

$x + \frac{1}{2} = 0$

Step 7.2

Subtract $\frac{1}{2}$ from both sides of the equation.

$x = - \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}$

Decimal Form:

$x = - 0.5$