How do you simplify $\sqrt[3]{192 x^{9}}$ $root3(192x^9)$ ?

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Answer 1 · 2015-09-11T09:53:19

$\sqrt[3]{192 x^{9}} = 4 \cdot x^{3} \cdot \sqrt[3]{3}$ $root3(192x^9) = color(green)(4 * x^3 *root3(3)$

In exponents, $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ $color(blue)(rootn(a*b) = rootna*rootnb$

Hence $\sqrt[3]{192 x^{9}} = \sqrt[3]{192} \cdot \sqrt[3]{x^{9}}$ $root3(192x^9) = root3(192)*root3(x^9)$ ---------(1)

Let's find $\sqrt[3]{192}$ $root3(192)$ first

$= \sqrt[3]{2^{6} \cdot 3}$ $= root3(2^6 * 3)$

$= \sqrt[3]{2^{6}} \cdot \sqrt[3]{3}$ $= root3(2^6)*root3(3)$

In exponents, $\sqrt[n]{a^{b}} = a^{\frac{b}{n}}$ $color(blue)(rootn(a^b) = a^(b/n)$

Hence $\sqrt[3]{2^{6}} \cdot \sqrt[3]{3} = 2^{\frac{6}{3}} \cdot \sqrt[3]{3}$ $root3(2^6)*root3(3) = 2^(6/3) *root3(3)$

$= 2^{2} \cdot \sqrt[3]{3}$ $= 2^2 *root3(3)$

$= 4 \cdot \sqrt[3]{3}$ $= 4 *root3(3)$

Now let's find $\sqrt[3]{x^{9}}$ $root3(x^9)$

$= x^{\frac{9}{3}}$ $= x^(9/3)$

$= x^{3}$ $= x^3$

Plugging the two values into (1), we get

$\sqrt[3]{192 x^{9}} = 4 \cdot \sqrt[3]{3} \cdot x^{3}$ $root3(192x^9) = 4 *root3(3)*x^3$

$\sqrt[3]{192 x^{9}} = 4 \cdot x^{3} \cdot \sqrt[3]{3}$ $root3(192x^9) = color(green)(4 * x^3 *root3(3)$

How do you simplify 3√192x9root3(192x^9)?