What is the following product?

[tex]\[ \sqrt{30} \cdot \sqrt{10} \][/tex]

A. [tex]\(2 \sqrt{10}\)[/tex]

B. [tex]\(3 \sqrt{10}\)[/tex]

C. [tex]\(4 \sqrt{10}\)[/tex]

D. [tex]\(10 \sqrt{3}\)[/tex]



Answer :

To solve the product [tex]\(\sqrt{30} \cdot \sqrt{10}\)[/tex], we start with the properties of square roots. Specifically, we use the property that the product of two square roots is the square root of the product of the two numbers:

[tex]\[ \sqrt{30} \cdot \sqrt{10} = \sqrt{30 \cdot 10} \][/tex]

Next, we calculate the product inside the square root:

[tex]\[ 30 \cdot 10 = 300 \][/tex]

Thus, the expression simplifies to:

[tex]\[ \sqrt{30} \cdot \sqrt{10} = \sqrt{300} \][/tex]

Now, we need to simplify [tex]\(\sqrt{300}\)[/tex]. Notice that 300 can be factored into:

[tex]\[ 300 = 100 \cdot 3 \][/tex]

Since the square root of a product is the product of the square roots:

[tex]\[ \sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} \][/tex]

We know that:

[tex]\[ \sqrt{100} = 10 \][/tex]

So, substituting back, we obtain:

[tex]\[ \sqrt{300} = 10 \cdot \sqrt{3} \][/tex]

The simplified form of the expression [tex]\(\sqrt{30} \cdot \sqrt{10}\)[/tex] is:

[tex]\[ \sqrt{30} \cdot \sqrt{10} = 10 \sqrt{3} \][/tex]

Thus, the correct answer is:
[tex]\[ 10 \sqrt{3} \][/tex]