To simplify [tex]\(\sqrt{75 x^{10}}\)[/tex], we will break it down into manageable steps.
1. Factor the number inside the square root:
[tex]\[
75 = 25 \times 3
\][/tex]
Therefore, we can rewrite [tex]\(\sqrt{75 x^{10}}\)[/tex] as:
[tex]\[
\sqrt{75 x^{10}} = \sqrt{25 \times 3 \times x^{10}}
\][/tex]
2. Use the property of square roots to split the expression:
[tex]\[
\sqrt{25 \times 3 \times x^{10}} = \sqrt{25} \times \sqrt{3} \times \sqrt{x^{10}}
\][/tex]
3. Evaluate each square root individually:
- [tex]\(\sqrt{25} = 5\)[/tex] because 25 is a perfect square.
- [tex]\(\sqrt{3}\)[/tex] remains [tex]\(\sqrt{3}\)[/tex] because it is an irrational number.
- For [tex]\(\sqrt{x^{10}}\)[/tex], we use the property of exponents and square roots: [tex]\(\sqrt{x^{10}} = (x^{10})^{1/2} = x^{10 \times \frac{1}{2}} = x^5\)[/tex].
4. Combine all parts:
[tex]\[
\sqrt{25} \times \sqrt{3} \times \sqrt{x^{10}} = 5 \times \sqrt{3} \times x^5
\][/tex]
Thus, the simplified form of [tex]\(\sqrt{75 x^{10}}\)[/tex] is:
[tex]\[
5 \sqrt{3} x^5
\][/tex]
