Simplify 9u^2 square root of 28uv^2+ square root of square root of 63u^5

$9 u^{2} \sqrt{28 u v^{2} + \sqrt{\sqrt{63 u^{5}}}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{63 u^{5}}$ as ${(63 u^{5})}^{\frac{1}{2}}$ .

$9 u^{2} \sqrt{28 u v^{2} + \sqrt{{(63 u^{5})}^{\frac{1}{2}}}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{{(63 u^{5})}^{\frac{1}{2}}}$ as ${(63 u^{5})}^{\frac{\frac{1}{2}}{2}}$ .

$9 u^{2} \sqrt{28 u v^{2} + {(63 u^{5})}^{\frac{\frac{1}{2}}{2}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$9 u^{2} \sqrt{28 u v^{2} + {(63 u^{5})}^{\frac{1}{2} \cdot \frac{1}{2}}}$

Step 4

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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

$9 u^{2} \sqrt{28 u v^{2} + {(63 u^{5})}^{\frac{1}{2 \cdot 2}}}$

Multiply $2$ by $2$ .

$9 u^{2} \sqrt{28 u v^{2} + {(63 u^{5})}^{\frac{1}{4}}}$

Step 5

Apply basic rules of exponents.

Apply the product rule to $63 u^{5}$ .

$9 u^{2} \sqrt{28 u v^{2} + 63^{\frac{1}{4}} {(u^{5})}^{\frac{1}{4}}}$

Multiply the exponents in ${(u^{5})}^{\frac{1}{4}}$ .

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

$9 u^{2} \sqrt{28 u v^{2} + 63^{\frac{1}{4}} u^{5 (\frac{1}{4})}}$

Combine $5$ and $\frac{1}{4}$ .

$9 u^{2} \sqrt{28 u v^{2} + 63^{\frac{1}{4}} u^{\frac{5}{4}}}$

Step 6

Factor $u$ out of $28 u v^{2} + 63^{\frac{1}{4}} u^{\frac{5}{4}}$ .

Factor $u$ out of $28 u v^{2}$ .

$9 u^{2} \sqrt{u (28 v^{2}) + 63^{\frac{1}{4}} u^{\frac{5}{4}}}$

Factor $u$ out of $63^{\frac{1}{4}} u^{\frac{5}{4}}$ .

$9 u^{2} \sqrt{u (28 v^{2}) + u (63^{\frac{1}{4}} u^{\frac{1}{4}})}$

Factor $u$ out of $u (28 v^{2}) + u (63^{\frac{1}{4}} u^{\frac{1}{4}})$ .

$9 u^{2} \sqrt{u (28 v^{2} + 63^{\frac{1}{4}} u^{\frac{1}{4}})}$