Question

I know the answers to these questions, I just can't seem to figure out how to find the answers. I'm not too good at math...

Simplify the radical exprssion

`√(14q) * 2 √(4q)`
Answer is
`4q √(14)`
Simplify the radical expression (There's supposed to be a big suare root sign over the whole fraction)

`( √(63x^15y^9) )/(7xy^11)` (^15 and ^11)
Answer is
`(3x^7)/(y)`
Simplify the radical expression

`2 √(6) + 3 √(96)`
Answer is
`14 √(6)`

Answer 1

`\bf √(14q)\cdot 2√(4q)\implies √(14q)\cdot 2√(2^2q)\implies √(14q)\cdot 2(2)√(q) \\\\\\ √(14q)\cdot 4√(q) \implies 4√(14q)\cdot √(q)\implies 4√(14q\cdot q)\implies 4√(14q^2) \\\\\\ 4q√(14)`

now, for the next one, bear in mind that, if you move a "factor" from the bottom to the top, the exponent changes sign, and if you move it from the top to the bottom, also changes sign, so, let's see,


`\bf \left.\qquad \qquad \right.\textit{negative exponents}\\\\ a^{-{ n}} \implies \cfrac{1}{a^( n)} \qquad \qquad \cfrac{1}{a^( n)}\implies a^{-{ n}} \qquad \qquad a^{{{ n}}}\implies \cfrac{1}{a^{-{{ n}}}}\\\\ -------------------------------\\\\ \sqrt{\cfrac{63x^(15)y^9}{7xy^(11)}}\implies \sqrt{\cfrac{63}{7}\cdot \cfrac{x^(15)y^9}{xy^(11)}}\implies \sqrt{ \cfrac{9x^(15)y^9}{xy^(11)}}\implies \sqrt{ \cfrac{9x^(15)x^(-1)}{y^(11)y^(-9)}}`


`\bf \sqrt{ \cfrac{9x^(15-1)}{y^(11-9)}}\implies \sqrt{\cfrac{9x^(14)}{y^2}}\implies \sqrt{\cfrac{9x^(7\cdot 2)}{y^2}}\implies \sqrt{\cfrac{3^2(x^7)^2}{y^2}}\implies \cfrac{3x^7}{y}\\\\ -------------------------------`


`\bf 2√(6)+3√(96)\qquad \begin{cases} 96=2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3\\ \qquad 2^2\cdot 2^2\cdot 2\cdot 3\\ \qquad (2\cdot 2)^2\cdot 2\cdot 3\\ \qquad 4^2\cdot 6 \end{cases}\implies 2√(6)+3√(4^2\cdot 6) \\\\\\ 2√(6)+3(4)√(6)\implies 2√(6)+12√(6)\implies 14√(6)`