To simplify the expression [tex]\(\sqrt{75 x^3} \cdot \sqrt{3 x^3}\)[/tex], let's go through the steps methodically:
1. Understand the properties of square roots:
Recall that [tex]\(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)[/tex]. This property allows us to combine the square root terms.
2. Apply the square root property:
Using this property, we can combine the square roots into a single square root:
[tex]\[
\sqrt{75 x^3} \cdot \sqrt{3 x^3} = \sqrt{(75 x^3) \cdot (3 x^3)}.
\][/tex]
3. Simplify inside the square root:
Now, we need to multiply the terms inside the square root:
[tex]\[
(75 x^3) \cdot (3 x^3) = 75 \cdot 3 \cdot x^3 \cdot x^3.
\][/tex]
Calculate the numerical part:
[tex]\[
75 \cdot 3 = 225.
\][/tex]
For the variable part, recall the exponent multiplication rule [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex]:
[tex]\[
x^3 \cdot x^3 = x^{3+3} = x^6.
\][/tex]
Combine these results:
[tex]\[
(75 x^3) \cdot (3 x^3) = 225 x^6.
\][/tex]
4. Taking the square root:
We now have the simplified expression inside the square root:
[tex]\[
\sqrt{225 x^6}.
\][/tex]
Use the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{225 x^6} = \sqrt{225} \cdot \sqrt{x^6}.
\][/tex]
Calculate the square root of each term:
[tex]\[
\sqrt{225} = 15,
\][/tex]
and for [tex]\(x^6\)[/tex], recall that [tex]\(\sqrt{x^6} = x^{6/2} = x^3\)[/tex].
5. Combine the results:
Therefore,
[tex]\[
\sqrt{225 x^6} = 15 \cdot x^3.
\][/tex]
6. Final expression:
The simplified expression for [tex]\(\sqrt{75 x^3} \cdot \sqrt{3 x^3}\)[/tex] is:
[tex]\[
15 x^3.
\][/tex]
Thus, the simplified form of the given expression is [tex]\(15 x^3\)[/tex].
