P. 3 Use the product rule to simplify the expressions in Exercises 41-44. In Exercises $$ 43-44 $$, assume that variables represent nonnegative real numbers. $$ \begin{array}{ll} ext { 41. } \sqrt{300} & ext { 42. } \sqrt{12 x^{2}} \ ext { 43. } \sqrt{10 x} \cdot \sqrt{2 x} & ext { 44. } \sqrt{r^{3}}\end{array} $$

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P. 3 Use the product rule to simplify the expressions in Exercises 41-44. In Exercises $$ 43-44 $$, assume that variables represent nonnegative real numbers. $$ \begin{array}{ll}	ext { 41. } \sqrt{300} & 	ext { 42. } \sqrt{12 x^{2}} \ 	ext { 43. } \sqrt{10 x} \cdot \sqrt{2 x} & 	ext { 44. } \sqrt{r^{3}}\end{array} $$

UpStudy AI · Accepted Answer

To simplify the expressions using the product rule for square roots, we can use the property that $$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$$.

Let's simplify each expression:

**41.** $$\sqrt{300}$$

To simplify $$\sqrt{300}$$, we can factor 300 into its prime factors:
$$
300 = 100 \cdot 3 = 10^2 \cdot 3
$$
Now, we can take the square root:
$$
\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10\sqrt{3}
$$

**42.** $$\sqrt{12x^2}$$

We can factor 12 and use the property of square roots:
$$
12 = 4 \cdot 3
$$
Thus,
$$
\sqrt{12x^2} = \sqrt{4 \cdot 3 \cdot x^2} = \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x^2} = 2\sqrt{3} \cdot x = 2x\sqrt{3}
$$

**43.** $$\sqrt{10x} \cdot \sqrt{2x}$$

Using the product rule:
$$
\sqrt{10x} \cdot \sqrt{2x} = \sqrt{(10x)(2x)} = \sqrt{20x^2}
$$
Now, we can simplify $$\sqrt{20x^2}$$:
$$
20 = 4 \cdot 5
$$
Thus,
$$
\sqrt{20x^2} = \sqrt{4 \cdot 5 \cdot x^2} = \sqrt{4} \cdot \sqrt{5} \cdot \sqrt{x^2} = 2\sqrt{5} \cdot x = 2x\sqrt{5}
$$

**44.** $$\sqrt{r^3}$$

We can rewrite $$r^3$$ as $$r^2 \cdot r$$:
$$
\sqrt{r^3} = \sqrt{r^2 \cdot r} = \sqrt{r^2} \cdot \sqrt{r} = r \cdot \sqrt{r} = r\sqrt{r}
$$

In summary, the simplified expressions are:
- **41.** $$10\sqrt{3}$$
- **42.** $$2x\sqrt{3}$$
- **43.** $$2x\sqrt{5}$$
- **44.** $$r\sqrt{r}$$
41. $$10\sqrt{3}$$ 42. $$2x\sqrt{3}$$ 43. $$2x\sqrt{5}$$ 44. $$r\sqrt{r}$$

P. 3
Use the product rule to simplify the expressions in Exercises 41-44.
In Exercises , assume that variables represent nonnegative
real numbers.

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P. 3 Use the product rule to simplify the expressions in Exercises 41-44. In Exercises , assume that variables represent nonnegative real numbers.