To simplify [tex]\(\sqrt{3} \cdot \sqrt{30}\)[/tex], we can use a property of square roots that states the product of two square roots is equal to the square root of the product of those two numbers. That is,
[tex]\[ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \][/tex]
Applying this property, we get:
[tex]\[ \sqrt{3} \cdot \sqrt{30} = \sqrt{3 \cdot 30} \][/tex]
Next, we need to simplify the expression inside the square root:
[tex]\[ 3 \cdot 30 = 90 \][/tex]
So we have:
[tex]\[ \sqrt{3} \cdot \sqrt{30} = \sqrt{90} \][/tex]
To simplify [tex]\(\sqrt{90}\)[/tex], we can factor 90 into its prime factors and look for perfect squares:
[tex]\[ 90 = 9 \times 10 = 3^2 \times 10 \][/tex]
We know that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], so we can write:
[tex]\[ \sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} \][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex], we get:
[tex]\[ \sqrt{90} = 3 \times \sqrt{10} \][/tex]
This can be simplified as:
[tex]\[ 3 \sqrt{10} \][/tex]
Therefore, the simplified form of [tex]\(\sqrt{3} \cdot \sqrt{30}\)[/tex] is:
[tex]\[ 3 \sqrt{10} \][/tex]
Numerically, we can calculate the value of [tex]\( 3 \sqrt{10} \)[/tex] as:
[tex]\[ 3 \cdot 3.162277660168379 \approx 9.486832980505138 \][/tex]
So, the simplified form of [tex]\(\sqrt{3} \cdot \sqrt{30}\)[/tex] is [tex]\( 3 \sqrt{10} \)[/tex], and its numerical value is approximately [tex]\( 9.486832980505138 \)[/tex].
