# How do you simplify sqrt(h^3)/sqrt8?

Dec 14, 2017

$\pm \frac{h \sqrt{2 h}}{4}$

#### Explanation:

There are two things to consider.

1) Are there any squared values we can 'take out' of the roots?

2) It is considered not good practice to have a root as or in the denominator so we need to change what is left of it into a whole number.

Given: $\frac{\sqrt{{h}^{3}}}{\sqrt{8}}$

Write as: $\frac{\sqrt{{h}^{2} \times h}}{\sqrt{{2}^{2} \times 2}} \textcolor{w h i t e}{\text{ddd")" giving: "color(white)("ddd}} \frac{h \sqrt{h}}{2 \sqrt{2}}$

Now we 'get rid of the root in the denominator; multiply by 1 and you do not change the inherent value. However 1 comes in many forms.

$\textcolor{g r e e n}{\frac{h \sqrt{h}}{2 \sqrt{2}} \textcolor{red}{\times 1} \textcolor{w h i t e}{\text{dddd")-> color(white)("dddd}} \frac{h \sqrt{h}}{2 \sqrt{2}} \textcolor{red}{\times \frac{\sqrt{2}}{\sqrt{2}}}}$

color(white)("d")color(green)(color(white)("ddddddddddd")->color(white)("dddd")(hsqrt(2h)) /4 )
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Foot note}}$

Compare the consequence of $\sqrt{h} \times \sqrt{2} \to \sqrt{2 h} \textcolor{w h i t e}{\text{d}}$ to the example:

$\sqrt{4} \times \sqrt{9} \textcolor{w h i t e}{\text{d")=color(white)("d}} 2 \times 3 = 6$

$\sqrt{4 \times 9} \textcolor{w h i t e}{\text{d")=color(white)("d")sqrt(36)color(white)("d")=color(white)("d}} 6$

color(brown)("So " hsqrth xxsqrt2color(white)("dd") =color(white)("dd") hsqrt(hxx2)color(white)("dd") =color(white)("dd") hsqrt(2h))

$\textcolor{w h i t e}{\text{d}}$

$\textcolor{b r o w n}{\text{and "2sqrt(2) xxsqrt2color(white)("d") =color(white)("d") 2sqrt(2xx2)color(white)("d") =color(white)("d") 2sqrt(4)color(white)("d") =color(white)("d}} 2 \times 2 = 4$

Technically the answer should be $\pm \frac{h \sqrt{2 h}}{4}$

As the square root of a number is $\pm$

Example:

$\left(- 2\right) \times \left(- 2\right) \textcolor{w h i t e}{\text{d")=color(white)("d")(+4)color(white)("d")=color(white)("d}} \left(+ 2\right) \times \left(+ 2\right)$