Calculus Examples

$\frac{2}{5} x^{2} \cdot \sqrt{x} = \frac{2}{5} x^{\frac{5}{2}}$

Step 1

Simplify both sides of the equation.

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Step 1.1

Combine $\frac{2}{5}$ and $x^{2}$ .

$\frac{2 x^{2}}{5} \cdot \sqrt{x} = \frac{2}{5} x^{\frac{5}{2}}$

Step 1.2

Combine $\frac{2 x^{2}}{5}$ and $\sqrt{x}$ .

$\frac{2 x^{2} \sqrt{x}}{5} = \frac{2}{5} x^{\frac{5}{2}}$

Step 1.3

Combine $\frac{2}{5}$ and $x^{\frac{5}{2}}$ .

$\frac{2 x^{2} \sqrt{x}}{5} = \frac{2 x^{\frac{5}{2}}}{5}$

Step 2

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

$\frac{2 x^{2} x^{\frac{1}{2}}}{5} = \frac{2 x^{\frac{5}{2}}}{5}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x^{2} x^{\frac{1}{2}} = 2 x^{\frac{5}{2}}$

Step 4

Multiply $x^{2}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.1

Move $x^{\frac{1}{2}}$ .

$2 (x^{\frac{1}{2}} x^{2}) = 2 x^{\frac{5}{2}}$

Step 4.2

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

$2 x^{\frac{1}{2} + 2} = 2 x^{\frac{5}{2}}$

Step 4.3

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

$2 x^{\frac{1}{2} + 2 \cdot \frac{2}{2}} = 2 x^{\frac{5}{2}}$

Step 4.4

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

$2 x^{\frac{1}{2} + \frac{2 \cdot 2}{2}} = 2 x^{\frac{5}{2}}$

Step 4.5

Combine the numerators over the common denominator.

$2 x^{\frac{1 + 2 \cdot 2}{2}} = 2 x^{\frac{5}{2}}$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $2$ by $2$ .

$2 x^{\frac{1 + 4}{2}} = 2 x^{\frac{5}{2}}$

Step 4.6.2

Add $1$ and $4$ .

$2 x^{\frac{5}{2}} = 2 x^{\frac{5}{2}}$

Step 5

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

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Step 5.1

Subtract $2 x^{\frac{5}{2}}$ from both sides of the equation.

$2 x^{\frac{5}{2}} - 2 x^{\frac{5}{2}} = 0$

Step 5.2

Subtract $2 x^{\frac{5}{2}}$ from $2 x^{\frac{5}{2}}$ .

$0 = 0$

Step 6

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 7

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$