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Chapter 10: Problem 86

Simplify each expression. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. $$\left(-2 x y^{2} \sqrt{3 x}\right)(x y \sqrt{6 x})$$

Short Answer

Expert verified
The simplified form of the expression is \( -6x^2y^3\sqrt{2} \).

Step by step solution

01

Break down the expression

Put the expression \( \left(-2 x y^{2} \sqrt{3 x}\right)(x y \sqrt{6 x}) \) into components to understand better.
02

Carry out the multiplication outside and inside the square root separately

Combine like terms using multiplication for those outside the square roots, and also combine the radicands by multiplication, this will result to: \( -2xy^3\sqrt{18x^2} \). Remember, the multiplication of terms outside the square root is totally separate from multiplication of terms inside the square root.
03

Simplify the square root

The square root \(\sqrt{18x^2}\) can be simplified further to \(3x\sqrt{2}\). This is achieved by factoring out get \( 18 = 2*3^2 \) and \( x^2 \) as \( x*x \). Then, factor out the perfect squares which results in \(3x\sqrt{2}\). Now our expression is transformed into \( -2xy^3*3x\sqrt{2} \).
04

Combine the multiplied parts

Now everything is simplified, multiply the non-square root part, which gives \( -6x^2y^3\sqrt{2} \). This is the simplest form.

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Most popular questions from this chapter

In Exercises \(39-64,\) rationalize each denominator. $$\sqrt[3]{\frac{3}{x y^{2}}}$$

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\frac{25}{\sqrt{5 x^{2} y}}$$

Let \(f(x)=x^{2} .\) Find \(f(\sqrt{a+1}-\sqrt{a-1})\)

In Exercises \(65-74,\) simplify each radical expression and then rationalize the denominator. $$\frac{12}{\sqrt[3]{-8 x^{5} y^{8}}}$$

In Exercises \(105-112,\) add or subtract as indicated. Begin by rationalizing denominators for all terms in which denominators contain radicals. $$\sqrt[3]{25}-\frac{15}{\sqrt[3]{5}}$$
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