Question

simplify.
\sqrt{x^{25}}
assume that the variable represents a positive real number.

Explanation:

Step1: Recall the property of square roots and exponents

For a positive real number \( x \) and a positive integer \( n \), \( \sqrt{x^n}=x^{\frac{n}{2}} \) when \( n \) is an integer. Here, we have \( \sqrt{x^{25}} \), so we can rewrite the square root as an exponent with base \( x \) and exponent \( \frac{25}{2} \).
\( \sqrt{x^{25}}=x^{\frac{25}{2}} \)

Step2: Convert the improper fraction exponent to a mixed number

We know that \( \frac{25}{2}=12+\frac{1}{2} \), so \( x^{\frac{25}{2}} = x^{12+\frac{1}{2}} \). Using the property of exponents \( a^{m + n}=a^m\times a^n \), we get \( x^{12+\frac{1}{2}}=x^{12}\times x^{\frac{1}{2}} \). And since \( x^{\frac{1}{2}}=\sqrt{x} \), we can also write this as \( x^{12}\sqrt{x} \). Another way is to note that when simplifying square roots, we can express the exponent as a quotient and remainder when divided by 2. Dividing 25 by 2, we get a quotient of 12 and a remainder of 1. So, \( \sqrt{x^{25}}=\sqrt{x^{2\times12 + 1}}=\sqrt{(x^{12})^2\times x} \). Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a,b\geq0 \)), we have \( \sqrt{(x^{12})^2\times x}=\sqrt{(x^{12})^2}\times\sqrt{x} \). Since \( x \) is a positive real number, \( \sqrt{(x^{12})^2}=x^{12} \). So, \( \sqrt{(x^{12})^2}\times\sqrt{x}=x^{12}\sqrt{x} \) or we can write \( x^{12.5} \) but the form with a mixed exponent or the product of a power and a square root is more standard. But also, using the exponent rule \( x^{\frac{m}{n}}=\sqrt[n]{x^m} \), we can think of \( \frac{25}{2}=12+\frac{1}{2} \), so \( x^{\frac{25}{2}} = x^{12}\times x^{\frac{1}{2}}=x^{12}\sqrt{x} \). Alternatively, we can write it as \( x^{12}\sqrt{x} \) or in the form of \( x^{12.5} \), but the simplified radical form is \( x^{12}\sqrt{x} \) or we can also express the exponent as a fraction: \( x^{\frac{25}{2}} \). But let's check the division of 25 by 2. 2 times 12 is 24, so 25 is 2*12 +1, so \( x^{25}=x^{2\times12 + 1}=(x^{12})^2\times x \), so taking the square root, we get \( x^{12}\sqrt{x} \).

Answer:

\( x^{12}\sqrt{x} \) (or \( x^{\frac{25}{2}} \))