What is $\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x}$ $sqrt(2x^3)sqrt(6x^2)sqrt(10x)$ in simplified form?

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What is $\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x}$ $sqrt(2x^3)sqrt(6x^2)sqrt(10x)$ in simplified form?

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Answer 1 · 2017-01-28T00:04:37

$\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x} = 2 x^{3} \sqrt{30}$ $sqrt(2x^3)*sqrt(6x^2)*sqrt(10x) = 2x^3sqrt(30)$

Explanation:

Note that if $a, b \geq 0$ $a, b >= 0$ then

$\sqrt{a} \sqrt{b} = \sqrt{a b}$ $sqrt(a)sqrt(b) = sqrt(ab)$

By extension, we find if $a, b, c \geq 0$ $a, b, c >= 0$ then:

$\sqrt{a} \sqrt{b} \sqrt{c} = \sqrt{a b} \sqrt{c} = \sqrt{a b c}$ $sqrt(a)sqrt(b)sqrt(c) = sqrt(ab)sqrt(c) = sqrt(abc)$

Note also that if $a \geq 0$ $a>=0$ then:

$\sqrt{a^{2}} = a$ $sqrt(a^2) = a$

In our example, we will assume $x \geq 0$ $x >= 0$ in order that all of the original square roots are well defined Real numbers.

So in our example:

$\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x} = \sqrt{2 x^{3} \cdot 6 x^{2} \cdot 10 x}$ $sqrt(2x^3)*sqrt(6x^2)*sqrt(10x) = sqrt(2x^3*6x^2*10x)$

$\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x} = \sqrt{2 x^{3} \cdot 2 x^{3} \cdot 30}$ $color(white)(sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)) = sqrt(2x^3*2x^3*30)$

$\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x} = \sqrt{{(2 x^{3})}^{2}} \sqrt{30}$ $color(white)(sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)) = sqrt((2x^3)^2)sqrt(30)$

$\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x} = 2 x^{3} \sqrt{30}$ $color(white)(sqrt(2x^3)*sqrt(6x^2)*sqrt(10x)) = 2x^3 sqrt(30)$

Footnote

It seems to be a common error to assume that:

$\sqrt{x^{2}} = x$ $sqrt(x^2) = x$

This does hold, but only if $x \geq 0$ $x >= 0$ .

If we want to cover the case $x < 0$ $x < 0$ too then we could write:

$\sqrt{x^{2}} = | x |$ $sqrt(x^2) = abs(x)$

In the given example, we can deduce that the case $x \geq 0$ $x >= 0$ is intended, since otherwise $\sqrt{2 x^{3}}$ $sqrt(2x^3)$ and $\sqrt{10 x}$ $sqrt(10x)$ could take imaginary values.

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What is $\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x}$ $sqrt(2x^3)sqrt(6x^2)sqrt(10x)$ in simplified form?

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What is $\sqrt{2 x^{3}} \cdot \sqrt{6 x^{2}} \cdot \sqrt{10 x}$ $sqrt(2x^3)sqrt(6x^2)sqrt(10x)$ in simplified form?