Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 6x⋅3x2\sqrt{6x}\cdot\sqrt{$$3x^2$$}

Accepted Answer

Identify the two functions being multiplied: here, the first function is $$\sqrt{6x}$$ and the second function is $$\sqrt{3x^2$}. $$Recall that the product rule for derivatives states: $$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$, but since the problem asks to simplify the product, focus on simplifying the expression $$\sqrt{6x} \cdot \sqrt{3x^2$}$$ first. Use the property of square roots that $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} to $$combine the two square roots into one: $$\sqrt{6x} \cdot \sqrt{3x^2$} = \sqrt{(6x)(3x^2$)}. $$Multiply inside the square root: $$(6x)(3x^2$$) = $$18x^3$$, so the expression becomes $$\sqrt{18x^3$}. $$Simplify $$\sqrt{18x^3$} by $$factoring inside the root to extract perfect squares: write $$18x^3 as$$ $$9 \cdot 2 \cdot x^2$ \cdot x$$, then use $$\sqrt{a^2$} = a to $$simplify.

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $\sqrt{6 x} \cdot \sqrt{3 x^{2}}$

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Key Concepts

Product Rule for Radicals

Properties of Exponents

Simplifying Radicals with Variables

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 6x⋅3x2\(\sqrt{6x}\[\cdot\]\sqrt{3x^2}\)