Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Cancel the common factor of $2^{x^{2}}$ and $2$ .

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Step 1.2.1

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

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

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Factor $2$ out of $2$ .

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

Step 1.2.2.2

Cancel the common factor.

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

Step 1.2.2.3

Rewrite the expression.

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

Step 1.2.2.4

Divide $2^{x^{2} - 1}$ by $1$ .

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

Step 2

Simplify each term.

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Step 2.1

Combine $3^{x^{2}}$ and $\frac{1}{3}$ .

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

Step 2.2

Cancel the common factor of $3^{x^{2}}$ and $3$ .

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $3$ out of $3$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.2.2.4

Divide $3^{x^{2} - 1}$ by $1$ .

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

Step 2.3

Multiply $- 2^{x^{2}} \cdot 4$ .

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Step 2.3.1

Multiply $4$ by $- 1$ .

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

Step 2.3.2

Factor out negative.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = - \sqrt{3}, \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = - \sqrt{3}, \sqrt{3}$

Decimal Form:

$x = - 1.73205080 \dots, 1.73205080 \dots$

Step 5