Question

a. choose the correct answer below
a. the square root is not completely removed. the radical \\(\sqrt{x}\\) remains in the expression after being rewritten
b. the square root is removed from the expression, and the simplified answer is x with an exponent of \\(\frac{n}{2}\\)
c. there is no pattern when n is even. the square root may or may not be removed from the expression after being rewritten
d. the square root is removed from the expression, and the simplified answer is x with an exponent of 2n
b. choose the correct answer below.
a. the square root is not completely removed. after being rewritten, the radical \\(\sqrt{x}\\) remains in the expression and is multiplied by x with an exponent of \\(\frac{n - 1}{2}\\) if n > 1.
b. there is no pattern when n is odd. the square root may or may not be removed from the expression after being rewritten.
c. the square root is not completely removed. the radical \\(\sqrt{x^{n - 1}}\\) remains in the expression after being rewritten.
d. the square root is removed from the expression, and the simplified answer is x with an exponent of \\(\frac{n - 1}{2}\\)

Explanation:

Part a

To simplify \(\sqrt{x^n}\) when \(n\) is even, we use the property of radicals \(\sqrt{a^m}=a^{\frac{m}{2}}\) (for \(a\geq0\)). For an even \(n\), \(x^n=(x^{\frac{n}{2}})^2\), so \(\sqrt{x^n}=x^{\frac{n}{2}}\). This removes the square root, and the exponent is \(\frac{n}{2}\). Let's analyze each option:

• Option A: Incorrect. The square root is removed as shown.
• Option B: Correct. As \(\sqrt{x^n}=x^{\frac{n}{2}}\) when \(n\) is even, the square root is removed, and the exponent is \(\frac{n}{2}\).
• Option C: Incorrect. There is a pattern (using the radical - exponent property) when \(n\) is even.
• Option D: Incorrect. The exponent should be \(\frac{n}{2}\), not \(2n\).

When \(n\) is odd, we can write \(x^n = x^{n - 1}\cdot x\), where \(n-1\) is even. Then \(\sqrt{x^n}=\sqrt{x^{n - 1}\cdot x}=\sqrt{x^{n - 1}}\cdot\sqrt{x}\). Since \(n - 1\) is even, \(\sqrt{x^{n - 1}}=x^{\frac{n - 1}{2}}\), so \(\sqrt{x^n}=x^{\frac{n - 1}{2}}\cdot\sqrt{x}\). This means the square root is not completely removed (the \(\sqrt{x}\) remains) and we have \(x\) with an exponent of \(\frac{n - 1}{2}\) multiplied by \(\sqrt{x}\) (for \(n>1\)). Let's analyze each option:

• Option A: Correct. This matches our derivation.
• Option B: Incorrect. There is a pattern (using the factorization of \(x^n\) for odd \(n\)).
• Option C: Incorrect. The radical remaining is \(\sqrt{x}\), not \(\sqrt{x^{n - 1}}\).
• Option D: Incorrect. The square root is not removed (we still have \(\sqrt{x}\) in the expression).

Answer:

B. The square root is removed from the expression, and the simplified answer is \(x\) with an exponent of \(\frac{n}{2}\)

Part b