Algebra Examples

$2^{x} = \frac{4^{x^{3}}}{8^{x^{2}}}$

Step 1

Move $8^{x^{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$2^{x} = 4^{x^{3}} \cdot 8^{- x^{2}}$

Step 2

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

$2^{x} = {(2^{2})}^{x^{3}} \cdot 8^{- x^{2}}$

Step 3

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

$2^{x} = 2^{2 x^{3}} \cdot 8^{- x^{2}}$

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Multiply $- 1$ by $3$ .

$2^{x} = 2^{2 x^{3}} \cdot 2^{- 3 x^{2}}$

Step 6

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

$2^{x} = 2^{2 x^{3} - 3 x^{2}}$

Step 7

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

$x = 2 x^{3} - 3 x^{2}$

Step 8

Solve for $x$ .

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Step 8.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{3} - 3 x^{2} = x$

Step 8.2

Subtract $x$ from both sides of the equation.

$2 x^{3} - 3 x^{2} - x = 0$

Step 8.3

Factor $x$ out of $2 x^{3} - 3 x^{2} - x$ .

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Step 8.3.1

Factor $x$ out of $2 x^{3}$ .

$x (2 x^{2}) - 3 x^{2} - x = 0$

Step 8.3.2

Factor $x$ out of $- 3 x^{2}$ .

$x (2 x^{2}) + x (- 3 x) - x = 0$

Step 8.3.3

Factor $x$ out of $- x$ .

$x (2 x^{2}) + x (- 3 x) + x \cdot - 1 = 0$

Step 8.3.4

Factor $x$ out of $x (2 x^{2}) + x (- 3 x)$ .

$x (2 x^{2} - 3 x) + x \cdot - 1 = 0$

Step 8.3.5

Factor $x$ out of $x (2 x^{2} - 3 x) + x \cdot - 1$ .

$x (2 x^{2} - 3 x - 1) = 0$

Step 8.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$2 x^{2} - 3 x - 1 = 0$

Step 8.5

Set $x$ equal to $0$ .

$x = 0$

Step 8.6

Set $2 x^{2} - 3 x - 1$ equal to $0$ and solve for $x$ .

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Step 8.6.1

Set $2 x^{2} - 3 x - 1$ equal to $0$ .

$2 x^{2} - 3 x - 1 = 0$

Step 8.6.2

Solve $2 x^{2} - 3 x - 1 = 0$ for $x$ .

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Step 8.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8.6.2.2

Substitute the values $a = 2$ , $b = - 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 8.6.2.3

Simplify.

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Step 8.6.2.3.1

Simplify the numerator.

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Step 8.6.2.3.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 8.6.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 8.6.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 8.6.2.3.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 8.6.2.3.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 8.6.2.3.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 8.6.2.4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 8.7

The final solution is all the values that make $x (2 x^{2} - 3 x - 1) = 0$ true.

$x = 0, \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Decimal Form:

$x = 0, 1.78077640 \dots, - 0.28077640 \dots$