Sure, let's simplify the expression [tex]\( 12 \sqrt{75} \)[/tex] step-by-step.
1. Factorize the number inside the square root:
The number 75 can be factorized into its prime factors.
[tex]\[
75 = 25 \times 3
\][/tex]
2. Use the property of square roots:
We know that [tex]\( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \)[/tex]. Applying this property here:
[tex]\[
\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}
\][/tex]
3. Simplify each square root:
We know that:
[tex]\[
\sqrt{25} = 5
\][/tex]
So:
[tex]\[
\sqrt{75} = 5 \times \sqrt{3}
\][/tex]
4. Multiply by the coefficient outside the square root:
We started with [tex]\( 12 \sqrt{75} \)[/tex]. Substituting [tex]\( \sqrt{75} \)[/tex] with [tex]\( 5 \sqrt{3} \)[/tex] gives:
[tex]\[
12 \sqrt{75} = 12 \times 5 \times \sqrt{3}
\][/tex]
5. Perform the multiplication:
[tex]\[
12 \times 5 = 60
\][/tex]
Therefore:
[tex]\[
12 \sqrt{75} = 60 \sqrt{3}
\][/tex]
So, the simplified form of the expression [tex]\( 12 \sqrt{75} \)[/tex] is:
[tex]\[
60 \sqrt{3}
\][/tex]
