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Which choice is equivalent to the product below?

[tex] \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6} [/tex]

A. [tex] 9 \sqrt{10} [/tex]
B. [tex] 3 \sqrt{5} [/tex]
C. [tex] 10 \sqrt{3} [/tex]
D. [tex] 3 \sqrt{10} [/tex]


Sagot :

Let's solve the given problem step-by-step.

We are given the expression:
[tex]\[ \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6} \][/tex]

First, let's recognize a property of square roots:
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]

Using this property, we can combine the square roots. Specifically:
[tex]\[ \sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6} = \sqrt{3 \cdot 5 \cdot 6} \][/tex]

Next, we calculate the product inside the square root:
[tex]\[ 3 \cdot 5 = 15 \][/tex]
[tex]\[ 15 \cdot 6 = 90 \][/tex]

So, the expression simplifies to:
[tex]\[ \sqrt{90} \][/tex]

Now, we need to determine which of the given choices is equivalent to [tex]\( \sqrt{90} \)[/tex].

Let's consider each option:

A. [tex]\( 9 \sqrt{10} \)[/tex]

To compare, we can consider the numerical simplification of [tex]\( 9 \sqrt{10} \)[/tex]:
[tex]\[ 9 \sqrt{10} = 9 \cdot \sqrt{10} \approx 28.4605 \][/tex]

This does not match [tex]\( \sqrt{90} \)[/tex].

B. [tex]\( 3 \sqrt{5} \)[/tex]

Comparing numerically:
[tex]\[ 3 \sqrt{5} = 3 \cdot \sqrt{5} \approx 6.7082 \][/tex]

This does not match [tex]\( \sqrt{90} \)[/tex].

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

Comparing numerically:
[tex]\[ 10 \sqrt{3} = 10 \cdot \sqrt{3} \approx 17.3205 \][/tex]

This does not match [tex]\( \sqrt{90} \)[/tex].

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

Comparing numerically:
[tex]\[ 3 \sqrt{10} = 3 \cdot \sqrt{10} \approx 9.4868 \][/tex]

This matches the value of [tex]\( \sqrt{90} \)[/tex], since:
[tex]\[ \sqrt{90} \approx 9.4868 \][/tex]

Therefore, the choice that is equivalent to the product [tex]\(\sqrt{3} \cdot \sqrt{5} \cdot \sqrt{6}\)[/tex] is:

[tex]\[ \boxed{3 \sqrt{10}} \][/tex]
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