Finite Math Examples

$3 x \sqrt{8 x} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x}$ as ${(8 x)}^{\frac{1}{2}}$ .

$3 x {(8 x)}^{\frac{1}{2}} - 4 \sqrt{x^{3}} + 3 x \sqrt{72 x}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$3 x {(8 x)}^{\frac{1}{2}} - 4 x^{\frac{3}{2}} + 3 x \sqrt{72 x}$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{72 x}$ as ${(72 x)}^{\frac{1}{2}}$ .

$3 x {(8 x)}^{\frac{1}{2}} - 4 x^{\frac{3}{2}} + 3 x {(72 x)}^{\frac{1}{2}}$

Step 4

Factor $x$ out of $3 x {(8 x)}^{\frac{1}{2}} - 4 x^{\frac{3}{2}} + 3 x {(72 x)}^{\frac{1}{2}}$ .

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Step 4.1

Factor $x$ out of $3 x {(8 x)}^{\frac{1}{2}}$ .

$x (3 {(8 x)}^{\frac{1}{2}}) - 4 x^{\frac{3}{2}} + 3 x {(72 x)}^{\frac{1}{2}}$

Step 4.2

Factor $x$ out of $- 4 x^{\frac{3}{2}}$ .

$x (3 {(8 x)}^{\frac{1}{2}}) + x (- 4 x^{\frac{1}{2}}) + 3 x {(72 x)}^{\frac{1}{2}}$

Step 4.3

Factor $x$ out of $3 x {(72 x)}^{\frac{1}{2}}$ .

$x (3 {(8 x)}^{\frac{1}{2}}) + x (- 4 x^{\frac{1}{2}}) + x (3 {(72 x)}^{\frac{1}{2}})$

Step 4.4

Factor $x$ out of $x (3 {(8 x)}^{\frac{1}{2}}) + x (- 4 x^{\frac{1}{2}})$ .

$x (3 {(8 x)}^{\frac{1}{2}} - 4 x^{\frac{1}{2}}) + x (3 {(72 x)}^{\frac{1}{2}})$

Step 4.5

Factor $x$ out of $x (3 {(8 x)}^{\frac{1}{2}} - 4 x^{\frac{1}{2}}) + x (3 {(72 x)}^{\frac{1}{2}})$ .

$x (3 {(8 x)}^{\frac{1}{2}} - 4 x^{\frac{1}{2}} + 3 {(72 x)}^{\frac{1}{2}})$

Step 5

Apply the product rule to $8 x$ .

$x (3 (8^{\frac{1}{2}} x^{\frac{1}{2}}) - 4 x^{\frac{1}{2}} + 3 {(72 x)}^{\frac{1}{2}})$

Step 6

Remove parentheses.

$x (3 \cdot 8^{\frac{1}{2}} x^{\frac{1}{2}} - 4 x^{\frac{1}{2}} + 3 {(72 x)}^{\frac{1}{2}})$

Step 7

Apply the product rule to $72 x$ .

$x (3 \cdot 8^{\frac{1}{2}} x^{\frac{1}{2}} - 4 x^{\frac{1}{2}} + 3 (72^{\frac{1}{2}} x^{\frac{1}{2}}))$

Step 8

Remove parentheses.

$x (3 \cdot 8^{\frac{1}{2}} x^{\frac{1}{2}} - 4 x^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} x^{\frac{1}{2}})$

Step 9

Reorder terms.

$x (3 \cdot 8^{\frac{1}{2}} x^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} x^{\frac{1}{2}} - 4 x^{\frac{1}{2}})$

Step 10

Factor.

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Step 10.1

Factor $x^{\frac{1}{2}}$ out of $3 \cdot 8^{\frac{1}{2}} x^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} x^{\frac{1}{2}} - 4 x^{\frac{1}{2}}$ .

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Step 10.1.1

Factor $x^{\frac{1}{2}}$ out of $3 \cdot 8^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$x (x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}}) + 3 \cdot 72^{\frac{1}{2}} x^{\frac{1}{2}} - 4 x^{\frac{1}{2}})$

Step 10.1.2

Factor $x^{\frac{1}{2}}$ out of $3 \cdot 72^{\frac{1}{2}} x^{\frac{1}{2}}$ .

$x (x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}}) + x^{\frac{1}{2}} (3 \cdot 72^{\frac{1}{2}}) - 4 x^{\frac{1}{2}})$

Step 10.1.3

Factor $x^{\frac{1}{2}}$ out of $- 4 x^{\frac{1}{2}}$ .

$x (x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}}) + x^{\frac{1}{2}} (3 \cdot 72^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 4)$

Step 10.1.4

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}}) + x^{\frac{1}{2}} (3 \cdot 72^{\frac{1}{2}})$ .

$x (x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 4)$

Step 10.1.5

Factor $x^{\frac{1}{2}}$ out of $x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}}) + x^{\frac{1}{2}} \cdot - 4$ .

$x (x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4))$

Step 10.2

Remove unnecessary parentheses.

$x \cdot x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4)$

Step 11

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

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Step 11.1

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

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Step 11.1.1

Raise $x$ to the power of $1$ .

$x^{1} x^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4)$

Step 11.1.2

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

$x^{1 + \frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4)$

Step 11.2

Write $1$ as a fraction with a common denominator.

$x^{\frac{2}{2} + \frac{1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4)$

Step 11.3

Combine the numerators over the common denominator.

$x^{\frac{2 + 1}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4)$

Step 11.4

Add $2$ and $1$ .

$x^{\frac{3}{2}} (3 \cdot 8^{\frac{1}{2}} + 3 \cdot 72^{\frac{1}{2}} - 4)$