What is the following product? Assume [tex]b \geq 0[/tex].

[tex] \sqrt{b} \cdot \sqrt{b} [/tex]

A. [tex] b \sqrt{b} [/tex]
B. [tex] 2 \sqrt{b} [/tex]
C. [tex] b [/tex]
D. [tex] b^2 [/tex]



Answer :

Certainly! Let's break down the given problem step-by-step.

We need to compute the product of [tex]\(\sqrt{b}\)[/tex] and [tex]\(\sqrt{b}\)[/tex], where [tex]\(b \geq 0\)[/tex].

### Step-by-Step Solution:

1. Express the problem in mathematical notation:

[tex]\[ \sqrt{b} \cdot \sqrt{b} \][/tex]

2. Recall the property of square roots:

The square root of a product is the product of the square roots. That is, if [tex]\( a \geq 0 \)[/tex] and [tex]\( b \geq 0 \)[/tex], then:

[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]

3. Apply this property to our expression:

Here, we have [tex]\(a = b\)[/tex] and [tex]\(b = b\)[/tex]:

[tex]\[ \sqrt{b} \cdot \sqrt{b} = \sqrt{b \cdot b} \][/tex]

4. Simplify the product inside the square root:

[tex]\[ b \cdot b = b^2 \][/tex]

5. Take the square root of the resulting expression:

[tex]\[ \sqrt{b^2} \][/tex]

6. Using the property of square roots, [tex]\(\sqrt{b^2} = b\)[/tex] for [tex]\(b \geq 0\)[/tex], we can simplify:

[tex]\[ \sqrt{b^2} = b \][/tex]

Therefore, the product [tex]\(\sqrt{b} \cdot \sqrt{b}\)[/tex] simplifies to [tex]\(b\)[/tex].

### Conclusion:

[tex]\[ \sqrt{b} \cdot \sqrt{b} = b \][/tex]

So, the answer to the given question is [tex]\(b\)[/tex].