Question

**#5.) express in simplest radical form: $sqrt{8} \times sqrt{6}$
#6.) simplify $\frac{4}{sqrt{5}}$
a) $\frac{4sqrt{5}}{5}$
b) $\frac{sqrt{5}}{4}$
c) $\frac{sqrt{6}}{3}$
d) $\frac{5sqrt{2}}{2}$
#7.) (sat question—systems of equations) which ordered pair satisfies the system of equations below? (show your work!)
$x = y - 3$
$\frac{x}{2}+2y = 6$
a. $(-3,0)$
b. $(0,3)$
c. $(6,-3)$
d. $(36,-6)$
*#8.) express in simplest radical form
$sqrt{4}+sqrt{120}-sqrt{9}+sqrt{270}$

Explanation:

#5 Step1: Combine radicals

$\sqrt{8} \times \sqrt{6} = \sqrt{8 \times 6}$

#5 Step2: Multiply inside radical

$\sqrt{48}$

#5 Step3: Factor out perfect square

$\sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}$

#5 Step4: Simplify perfect square

$4\sqrt{3}$

#6 Step1: Rationalize the denominator

$\frac{4}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}$

#6 Step2: Multiply numerator/denominator

$\frac{4\sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{4\sqrt{5}}{5}$

#7 Step1: Substitute $x=y-3$

$\frac{y-3}{2} + 2y = 6$

#7 Step2: Eliminate denominator

$(y-3) + 4y = 12$

#7 Step3: Combine like terms

$5y - 3 = 12$

#7 Step4: Solve for $y$

$5y = 15 \implies y=3$

#7 Step5: Find $x$

$x = 3 - 3 = 0$

#7 Step6: Verify ordered pair

Check $(0,3)$ in both equations:
$0=3-3$ (true), $\frac{0}{2}+2(3)=6$ (true)

#8 Step1: Simplify each radical

$\sqrt{4}=2$, $\sqrt{120}=\sqrt{4\times30}=2\sqrt{30}$, $\sqrt{9}=3$, $\sqrt{270}=\sqrt{9\times30}=3\sqrt{30}$

#8 Step2: Combine like terms

$2 + 2\sqrt{30} - 3 + 3\sqrt{30}$

#8 Step3: Simplify constants and radicals

$(2-3) + (2\sqrt{30}+3\sqrt{30}) = -1 + 5\sqrt{30}$

Answer:

#5: $4\sqrt{3}$
#6: A) $\frac{4\sqrt{5}}{5}$
#7: b. $(0, 3)$
#8: $5\sqrt{30} - 1$