Question

suppose the combined length of the body and neck of a guitar is given by the expression ( 8sqrt{147} ). which radical expression represents the length of the body if the neck measures ( 8sqrt{48} )?
a ( 24sqrt{3} )
b ( 12sqrt{3} )
c ( 8sqrt{3} )
d ( sqrt{99} )

Explanation:

Step1: Recall the subtraction of radical expressions

To find the length of the body, we subtract the length of the neck from the combined length. So we need to compute \( 8\sqrt{147}-8\sqrt{48} \).

Step2: Simplify \( \sqrt{147} \)

Factor 147: \( 147 = 49\times3 \). Then \( \sqrt{147}=\sqrt{49\times3}=\sqrt{49}\times\sqrt{3}=7\sqrt{3} \). So \( 8\sqrt{147}=8\times7\sqrt{3}=56\sqrt{3} \).

Step3: Simplify \( \sqrt{48} \)

Factor 48: \( 48 = 16\times3 \). Then \( \sqrt{48}=\sqrt{16\times3}=\sqrt{16}\times\sqrt{3}=4\sqrt{3} \). So \( 8\sqrt{48}=8\times4\sqrt{3}=32\sqrt{3} \).

Step4: Subtract the two simplified expressions

\( 8\sqrt{147}-8\sqrt{48}=56\sqrt{3}-32\sqrt{3}=(56 - 32)\sqrt{3}=24\sqrt{3} \)? Wait, no, wait, maybe I made a mistake. Wait, wait, the combined length is \( 8\sqrt{147} \)? Wait, no, the problem says "the combined length of the body and neck of a guitar is given by the expression \( 8\sqrt{147} \). Which radical expression represents the length of the body if the neck measures \( 8\sqrt{48} \)?" Wait, maybe I misread. Wait, let's re - check.

Wait, maybe the combined length is \( 8\sqrt{147} \)? No, wait, the original problem: "Suppose the combined length of the body and neck of a guitar is given by the expression \( 8\sqrt{147} \). Which radical expression represents the length of the body if the neck measures \( 8\sqrt{48} \)?"

Wait, let's re - simplify:

Simplify \( \sqrt{147} \): \( 147 = 49\times3 \), so \( \sqrt{147}=7\sqrt{3} \), so \( 8\sqrt{147}=8\times7\sqrt{3}=56\sqrt{3} \)

Simplify \( \sqrt{48} \): \( 48 = 16\times3 \), so \( \sqrt{48}=4\sqrt{3} \), so \( 8\sqrt{48}=8\times4\sqrt{3}=32\sqrt{3} \)

Then body length = combined length - neck length = \( 56\sqrt{3}-32\sqrt{3}=(56 - 32)\sqrt{3}=24\sqrt{3} \)? But the option A is \( 24\sqrt{3} \). Wait, but let's check again. Wait, maybe the combined length is \( 8\sqrt{147} \) or is it \( 8\sqrt{147} \) written as \( 8\sqrt{147} \)? Wait, maybe I made a mistake in the problem statement. Wait, the user's image: "Suppose the combined length of the body and neck of a guitar is given by the expression \( 8\sqrt{147} \). Which radical expression represents the length of the body if the neck measures \( 8\sqrt{48} \)?"

Wait, let's do the subtraction again:

\( 8\sqrt{147}-8\sqrt{48}=8(\sqrt{147}-\sqrt{48}) \)

Simplify \( \sqrt{147}=\sqrt{49\times3}=7\sqrt{3} \), \( \sqrt{48}=\sqrt{16\times3}=4\sqrt{3} \)

So \( \sqrt{147}-\sqrt{48}=7\sqrt{3}-4\sqrt{3}=3\sqrt{3} \)

Then \( 8\times3\sqrt{3}=24\sqrt{3} \). Yes, that's correct.

Answer:

A. \( 24\sqrt{3} \)