Basic Math Examples

$\sqrt{a b} = \sqrt{a} \sqrt{b}$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{a b}}^{2} = {(\sqrt{a} \sqrt{b})}^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

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

${({(a b)}^{\frac{1}{2}})}^{2} = {(\sqrt{a} \sqrt{b})}^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(a b)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(a b)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(a b)}^{\frac{1}{2} \cdot 2} = {(\sqrt{a} \sqrt{b})}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(a b)}^{\frac{1}{2} \cdot 2} = {(\sqrt{a} \sqrt{b})}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(a b)}^{1} = {(\sqrt{a} \sqrt{b})}^{2}$

Step 2.2.1.2

Simplify.

$a b = {(\sqrt{a} \sqrt{b})}^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify ${(\sqrt{a} \sqrt{b})}^{2}$ .

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Step 2.3.1.1

Combine using the product rule for radicals.

$a b = {\sqrt{a b}}^{2}$

Step 2.3.1.2

Rewrite ${\sqrt{a b}}^{2}$ as $a b$ .

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Step 2.3.1.2.1

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

$a b = {({(a b)}^{\frac{1}{2}})}^{2}$

Step 2.3.1.2.2

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

$a b = {(a b)}^{\frac{1}{2} \cdot 2}$

Step 2.3.1.2.3

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

$a b = {(a b)}^{\frac{2}{2}}$

Step 2.3.1.2.4

Cancel the common factor of $2$ .

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Step 2.3.1.2.4.1

Cancel the common factor.

$a b = {(a b)}^{\frac{2}{2}}$

Step 2.3.1.2.4.2

Rewrite the expression.

$a b = {(a b)}^{1}$

Step 2.3.1.2.5

Simplify.

$a b = a b$

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

Subtract $a b$ from both sides of the equation.

$a b - a b = 0$

Step 3.1.2

Subtract $a b$ from $a b$ .

$0 = 0$

Step 3.2

Since $0 = 0$ , the equation will always be true.

Always true

Step 4

The result can be shown in multiple forms.

Always true

Interval Notation:

$(- \infty, \infty)$