Question

Please answer the following questions

Answer 1

Answer:

1 - c

2 - c

3 - b

Step-by-step explanation:

1.

$\begin{aligned}\sqrt{12}\times\sqrt{15}&=\sqrt{12\times15}\\&=\sqrt{180}\\&=\sqrt{36\times5}\\&=\sqrt{36}\times\sqrt{5}\\&=6\sqrt5 \end{aligned}$

Thus, the correct answer is option c.

2.

a.

$\begin{aligned}\sqrt[12]{(x^4)^\frac{1}{3}}&=\left((x^4)^\frac{1}{3} \right )^\frac{1}{12}\\&=x^\frac{1}{9} \end{aligned}$

b.

$\begin{aligned}x^\frac{12}{7}\times x^\frac{7}{12}&=x^{\frac{12}{7}+\frac{7}{12}}\\&=x^\frac{144+49}{84}\\&=x^\frac{193}{84}\end{aligned}$

c.

$\begin{aligned}\left(\sqrt{x^3} \right )^\frac{2}{3}&=\left((x^3)^\frac{1}{2} \right )^\frac{2}{3}\\&=x^{3\times\frac{1}{2}\times\frac{2}{3}}\\&=x\end{aligned}$

d.

$\begin{aligned}x^\frac{12}{7}-x^\frac{5}{7}&=x^\frac{5}{7}\left(x-1\right)\end{aligned}$

Thus, the correct answer is option c.

3.

Given: $x=3+\sqrt8$

Calculate $x^2\, \rm{and} \,\frac{1}{x^2}$.

$\begin{aligned}x^2&=(3+\sqrt8)^2\\&=3^2+2\times3\times\sqrt8+(\sqrt8)^2\\&=9+6\sqrt8+8\\&=17+6\sqrt8\end{aligned}$

$\begin{aligned}\frac{1}{x^2}&=\frac{1}{17+6\sqrt8}\\&=\frac{1}{17+6\sqrt8}\times\frac{17-6\sqrt8}{17-6\sqrt8}\\&=\frac{17-6\sqrt8}{(17)^2-(6\sqrt8)^2}\\&=\frac{17-6\sqrt8}{289-288}\\&=17-6\sqrt8\end{aligned}$

Calculate $x^2+\frac{1}{x^2}$.

$\begin{aligned}x^2+\frac{1}{x^2}&=(17+6\sqrt8)+(17-6\sqrt8)\\&=34\end{aligned}$

Thus, the correct answer is option b.

Answer 2

Answer:

i think you have not cropped your question properly