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QuestionSimplify −6x150x3+624x5-6 x \sqrt{150 x^{3}} + 6 \sqrt{24 x^{5}} to its simplest radical form.

Studdy Solution

STEP 1

Assumptions1. The given expression is −6x150x3+624x5-6 x \sqrt{150 x^{3}}+6 \sqrt{24 x^{5}} . We need to simplify this expression to its simplest radical form

STEP 2

First, we need to simplify the radicals. We can do this by factoring out the perfect squares from under the square root.

STEP 3

Let's start with the first term, 150x3\sqrt{150 x^{3}}. We can factor out the perfect squares as follows150x3=25×6×x2×x=5x6x\sqrt{150 x^{3}} = \sqrt{25 \times6 \times x^{2} \times x} =5x\sqrt{6x}

STEP 4

Now, let's simplify the second term, 24x\sqrt{24 x^{}}. We can factor out the perfect squares as follows24x=4×6×x4×x=2x26x\sqrt{24 x^{}} = \sqrt{4 \times6 \times x^{4} \times x} =2x^{2}\sqrt{6x}

STEP 5

Now, substitute these simplified radicals back into the original expression−x150x3+24x5=−x⋅5xx+⋅2x2x- x \sqrt{150 x^{3}}+ \sqrt{24 x^{5}} = -x \cdot5x\sqrt{x} + \cdot2x^{2}\sqrt{x}

STEP 6

implify the coefficients and the xx terms−6x⋅5x6x+6⋅2x26x=−30x26x+12x26x-6x \cdot5x\sqrt{6x} +6 \cdot2x^{2}\sqrt{6x} = -30x^{2}\sqrt{6x} +12x^{2}\sqrt{6x}

STEP 7

Now, we can combine like terms−30x26x+12x26x=(−30+12)x26x-30x^{2}\sqrt{6x} +12x^{2}\sqrt{6x} = (-30 +12)x^{2}\sqrt{6x}

STEP 8

implify the coefficients(−30+12)x26x=−18x26x(-30 +12)x^{2}\sqrt{6x} = -18x^{2}\sqrt{6x}The simplified expression in its simplest radical form is −18x26x-18x^{2}\sqrt{6x}.

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