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QuestionSimplify −x3x5−3x23x3-x \sqrt{3 x^{5}}-3 x^{2} \sqrt{3 x^{3}} to its simplest radical form.

Studdy Solution

STEP 1

Assumptions1. We are given the expression −x3x5−3x3x3-x \sqrt{3 x^{5}}-3 x^{} \sqrt{3 x^{3}} . We are asked to simplify this expression to its simplest radical form3. We assume that xx is a real number

STEP 2

First, we can rewrite the square roots as fractional powers.−xx5−x2x=−x(x5)1/2−x2(x)1/2-x \sqrt{ x^{5}}- x^{2} \sqrt{ x^{}} = -x (x^{5})^{1/2} -x^{2} (x^{})^{1/2}

STEP 3

Apply the power of a power rule, which states that (am)n=amn(a^m)^n = a^{mn}, to simplify the expression.
−x(3x5)1/2−3x2(3x3)1/2=−x(31/2x5/2)−3x2(31/2x3/2)-x (3x^{5})^{1/2} -3x^{2} (3x^{3})^{1/2} = -x (3^{1/2}x^{5/2}) -3x^{2} (3^{1/2}x^{3/2})

STEP 4

Now, we can distribute the −x-x and −3x2-3x^{2} into the parentheses.
−x(31/2x/2)−3x2(31/2x3/2)=−x31/2x/2−3x231/2x3/2-x (3^{1/2}x^{/2}) -3x^{2} (3^{1/2}x^{3/2}) = -x3^{1/2}x^{/2} -3x^{2}3^{1/2}x^{3/2}

STEP 5

implify the expression by combining like terms.
−x31/2x5/2−3x231/2x3/2=−31/2x7/2−33/2x7/2-x3^{1/2}x^{5/2} -3x^{2}3^{1/2}x^{3/2} = -3^{1/2}x^{7/2} -3^{3/2}x^{7/2}

STEP 6

Now, we can factor out the common factor of −31/2x/2-3^{1/2}x^{/2}.
−31/2x/2−33/2x/2=−31/2x/2(1+3)-3^{1/2}x^{/2} -3^{3/2}x^{/2} = -3^{1/2}x^{/2}(1 +3)

STEP 7

implify the expression inside the parentheses.
−31/2x7/2(1+3)=−31/2x7/2(4)-3^{1/2}x^{7/2}(1 +3) = -3^{1/2}x^{7/2}(4)

STEP 8

Finally, simplify the expression to its simplest radical form.
−31/2x7/2(4)=−43x7/2-3^{1/2}x^{7/2}(4) = -4\sqrt{3}x^{7/2}The simplified expression is −43x7/2-4\sqrt{3}x^{7/2}.

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