Pre-Algebra Examples

$\sqrt{3 a b (\sqrt{3 a + \sqrt{9 b}})}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{9 b}$ as ${(9 b)}^{\frac{1}{2}}$ .

$\sqrt{3 a b \sqrt{3 a + {(9 b)}^{\frac{1}{2}}}}$

Step 2

Apply the product rule to $9 b$ .

$\sqrt{3 a b \sqrt{3 a + 9^{\frac{1}{2}} b^{\frac{1}{2}}}}$

Step 3

Rewrite $9$ as $3^{2}$ .

$\sqrt{3 a b \sqrt{3 a + {(3^{2})}^{\frac{1}{2}} b^{\frac{1}{2}}}}$

Step 4

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

$\sqrt{3 a b \sqrt{3 a + 3^{2 (\frac{1}{2})} b^{\frac{1}{2}}}}$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$\sqrt{3 a b \sqrt{3 a + 3^{2 (\frac{1}{2})} b^{\frac{1}{2}}}}$

Step 5.2

Rewrite the expression.

$\sqrt{3 a b \sqrt{3 a + 3^{1} b^{\frac{1}{2}}}}$

Step 6

Evaluate the exponent.

$\sqrt{3 a b \sqrt{3 a + 3 b^{\frac{1}{2}}}}$

Step 7

Factor $3$ out of $3 a + 3 b^{\frac{1}{2}}$ .

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Step 7.1

Factor $3$ out of $3 a$ .

$\sqrt{3 a b \sqrt{3 (a) + 3 b^{\frac{1}{2}}}}$

Step 7.2

Factor $3$ out of $3 b^{\frac{1}{2}}$ .

$\sqrt{3 a b \sqrt{3 (a) + 3 (b^{\frac{1}{2}})}}$

Step 7.3

Factor $3$ out of $3 (a) + 3 (b^{\frac{1}{2}})$ .

$\sqrt{3 a b \sqrt{3 (a + b^{\frac{1}{2}})}}$