QuestionSimplify the expression: $-9 x \sqrt{96 x^{3}} + x^{2} \sqrt{6 x}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the expression $-9 x \sqrt{96 x^{3}}+x^{} \sqrt{6 x}$ . We need to simplify this expression into simplest radical form.

STEP 2

First, we need to simplify the terms under the square roots. We can do this by factoring out the greatest perfect square from each term.

STEP 3

Factor out the greatest perfect square from $96x^{3}$ . The greatest perfect square factor of $96x^{3}$ is $16x^{2}$ . So, we can write $96x^{3}$ as $16x^{2} \cdot6x$ .

STEP 4

Substitute $16x^{2} \cdot6x$ for $96x^{3}$ in the first term of the expression.
$-9 x \sqrt{16x^{2} \cdot6x}$

STEP 5

Factor out the greatest perfect square from $x$ . The greatest perfect square factor of $x$ is $x$ . So, we can write $x$ as $x \cdot$ .

STEP 6

Substitute $x \cdot6$ for $6x$ in the second term of the expression.
$x^{2} \sqrt{x \cdot6}$

STEP 7

Now, we can simplify the square roots in both terms. The square root of $16x^{2}$ is $4x$ and the square root of $x$ is $x$ .

STEP 8

Substitute $4x$ for $\sqrt{16x^{2}}$ and $x$ for $\sqrt{x}$ in the expression.
$- x \cdot4x \sqrt{6x} + x^{2} \cdot x \sqrt{6}$

STEP 9

implify the expression by multiplying the terms outside the square roots.
$-36x^{2} \sqrt{6x} + x^{3} \sqrt{6}$

STEP 10

The expression $-36x^{2} \sqrt{6x} + x^{3} \sqrt{6}$ is the simplest radical form of the given expression.
So, $-9 x \sqrt{96 x^{3}}+x^{2} \sqrt{6 x}$ simplifies to $-36x^{2} \sqrt{6x} + x^{3} \sqrt{6}$ .

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