# How do you simplify sqrt[81x^12y^8z^10]?

Sep 9, 2015

$\sqrt{81 {x}^{12} {y}^{8} {z}^{10}} = 9 {x}^{6} {y}^{4} {z}^{5}$

#### Explanation:

You can use the following rule:

$\sqrt{a b} = \sqrt{a} \times \sqrt{b}$

In order to see how to simplify $\sqrt{81 {x}^{12} {y}^{8} {z}^{10}}$, we can split it as follows:

$\sqrt{81 {x}^{12} {y}^{8} {z}^{10}}$
$= \sqrt{81} \times \sqrt{{x}^{12}} \times \sqrt{{y}^{8}} \times \sqrt{{z}^{10}}$

Remember that $\sqrt{a} = {a}^{\frac{1}{2}}$, so:
$\sqrt{{a}^{n}}$
$= {a}^{{n}^{\frac{1}{2}}}$
$= {a}^{\frac{n}{2}}$

In other words, square rooting an expression with an exponent halves the exponent.

$\sqrt{81} \times \sqrt{{x}^{12}} \times \sqrt{{y}^{8}} \times \sqrt{{z}^{10}}$
$= 9 \times {x}^{6} \times {y}^{4} \times {z}^{5}$

$= 9 {x}^{6} {y}^{4} {z}^{5}$

Note: usually teachers will be fine with you skipping straight from $\sqrt{81 {x}^{12} {y}^{8} {z}^{10}}$ to $9 {x}^{6} {y}^{4} {z}^{5}$. You don't need to show the long process all the time unless your teacher asks for it.

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Sep 9, 2015

 sqrt[81x^12y^8z^10] = color(green)(9*x^(6)*y^(4)*z^(5)

#### Explanation:

There are two simple Exponents rules we need to know to answer this question

1) color(blue)(sqrt(a*b*c) = sqrta*sqrtb*sqrtc

2) color(blue)(sqrt(a^m) = a^(m/2)

Based on the first rule,

$\sqrt{81 {x}^{12} {y}^{8} {z}^{10}} = \sqrt{81} \cdot \sqrt{{x}^{12}} \cdot \sqrt{{y}^{8}} \cdot \sqrt{{z}^{10}}$

$= \sqrt{{9}^{2}} \cdot \sqrt{{x}^{12}} \cdot \sqrt{{y}^{8}} \cdot \sqrt{{z}^{10}}$

Based on the second rule, we write the above expression as

$= {9}^{\frac{2}{2}} \cdot {x}^{\frac{12}{2}} \cdot {y}^{\frac{8}{2}} \cdot {z}^{\frac{10}{2}}$

 = color(green)(9*x^(6)*y^(4)*z^(5)