Question

4
select the correct answer.
consider this expression.
\\(\frac{x - 4}{2(x + 4)}\\)
for which product or quotient is this expression the simplest form?
\\(\bigcirc\\) a. \\(\frac{x^2 - 16}{2x + 8} \div \frac{x + 4}{x^2 + 8x + 16}\\)
\\(\bigcirc\\) b. \\(\frac{2x + 8}{x^2 - 16} \div \frac{x^2 + 8x + 16}{x + 4}\\)
\\(\bigcirc\\) c. \\(\frac{x^2 - 16}{2x + 8} \cdot \frac{x + 4}{x^2 + 8x + 16}\\)
\\(\bigcirc\\) d. \\(\frac{2x + 8}{x^2 - 16} \cdot \frac{x^2 + 8x + 16}{x + 4}\\)

Explanation:

Step1: Factor all expressions

First, factor each quadratic and linear expression. Recall that \(a^2 - b^2=(a - b)(a + b)\) and \(a^2+2ab + b^2=(a + b)^2\).
- \(x^2 - 16=(x - 4)(x + 4)\)
- \(2x + 8 = 2(x + 4)\)
- \(x^2+8x + 16=(x + 4)^2\)

Step2: Simplify Option A

Option A: \(\frac{x^2 - 16}{2x + 8}\div\frac{x + 4}{x^2+8x + 16}\)
Dividing by a fraction is multiplying by its reciprocal: \(\frac{x^2 - 16}{2x + 8}\times\frac{x^2+8x + 16}{x + 4}\)
Substitute factored forms: \(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{(x + 4)^2}{x + 4}\)
Cancel common factors: \(\frac{(x - 4)\cancel{(x + 4)}}{2\cancel{(x + 4)}}\times\frac{\cancel{(x + 4)^2}}{\cancel{x + 4}}=\frac{x - 4}{2}\times1=\frac{x - 4}{2(x + 4)}\)? Wait, no, wait. Wait, let's re - do the cancellation.
Wait, \(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{(x + 4)^2}{x + 4}\):
- Cancel \((x + 4)\) in the numerator and denominator of the first fraction: \(\frac{(x - 4)}{2}\)
- Cancel \((x + 4)^2\) and \((x + 4)\) in the second fraction: \(\frac{(x + 4)^2}{x + 4}=x + 4\)
Wait, no, I made a mistake. Let's do it step by step.
\(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{(x + 4)^2}{x + 4}\)
First fraction: \(\frac{(x - 4)\cancel{(x + 4)}}{2\cancel{(x + 4)}}=\frac{x - 4}{2}\)
Second fraction: \(\frac{\cancel{(x + 4)^2}}{\cancel{x + 4}}=x + 4\)
Now multiply them: \(\frac{x - 4}{2}\times(x + 4)=\frac{(x - 4)(x + 4)}{2}\)? No, that's not right. Wait, no, the original division is \(\frac{x^2 - 16}{2x + 8}\div\frac{x + 4}{x^2+8x + 16}=\frac{x^2 - 16}{2x + 8}\times\frac{x^2+8x + 16}{x + 4}\)
\(x^2 - 16=(x - 4)(x + 4)\), \(2x + 8 = 2(x + 4)\), \(x^2+8x + 16=(x + 4)^2\)
So substitute: \(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{(x + 4)^2}{x + 4}\)
Cancel \((x + 4)\) from the first fraction's numerator and denominator: \(\frac{x - 4}{2}\)
Cancel \((x + 4)^2\) and \((x + 4)\) in the second fraction: \(\frac{(x + 4)^2}{x + 4}=x + 4\)
Now multiply \(\frac{x - 4}{2}\times(x + 4)\)? No, wait, no. Wait, the second fraction is \(\frac{(x + 4)^2}{x + 4}=(x + 4)\), and the first fraction is \(\frac{(x - 4)(x + 4)}{2(x + 4)}=\frac{x - 4}{2}\)
So multiplying \(\frac{x - 4}{2}\times(x + 4)\) is wrong. Wait, I messed up the reciprocal. Wait, division of fractions: \(a\div b=a\times\frac{1}{b}\). So \(\frac{x^2 - 16}{2x + 8}\div\frac{x + 4}{x^2+8x + 16}=\frac{x^2 - 16}{2x + 8}\times\frac{x^2+8x + 16}{x + 4}\)
Now, let's cancel correctly:
- Numerator of first fraction: \((x - 4)(x + 4)\)
- Denominator of first fraction: \(2(x + 4)\)
- Numerator of second fraction: \((x + 4)^2\)
- Denominator of second fraction: \(x + 4\)

So, cancel \((x + 4)\) from \((x - 4)(x + 4)\) and \(2(x + 4)\): we get \(\frac{x - 4}{2}\)
Cancel \((x + 4)^2\) and \(x + 4\): we get \(x + 4\)
Now multiply \(\frac{x - 4}{2}\times(x + 4)\)? No, that's not. Wait, no, the second fraction's numerator is \((x + 4)^2\) and denominator is \(x + 4\), so \(\frac{(x + 4)^2}{x + 4}=x + 4\)
The first fraction: \(\frac{(x - 4)(x + 4)}{2(x + 4)}=\frac{x - 4}{2}\)
Now multiply \(\frac{x - 4}{2}\times(x + 4)=\frac{(x - 4)(x + 4)}{2}\)? No, that's not the target expression. Wait, maybe I made a mistake in option A. Let's check option B.

Step3: Simplify Option B

Option B: \(\frac{2x + 8}{x^2 - 16}\div\frac{x^2+8x + 16}{x + 4}\)
Reciprocal of the second fraction: \(\frac{2x + 8}{x^2 - 16}\times\frac{x + 4}{x^2+8x + 16}\)
Substitute factored forms: \(\frac{2(x + 4)}{(x - 4)(x + 4)}\times\frac{x + 4}{(x + 4)^2}\)
Cancel common factors:
- Cancel \((x + 4)\) in the numerator and denominator of the first fraction: \(\frac{2}{(x - 4)}\)
- Cancel \((x + 4)\) in the numerator and denominator of the second fraction: \(\frac{\cancel{x + 4}}{\cancel{(x + 4)^2}}=\frac{1}{x + 4}\)
Now multiply: \(\frac{2}{x - 4}\times\frac{1}{x + 4}=\frac{2}{(x - 4)(x + 4)}\), which is not the target.

Step4: Simplify Option C

Option C: \(\frac{x^2 - 16}{2x + 8}\times\frac{x + 4}{x^2+8x + 16}\)
Substitute factored forms: \(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{x + 4}{(x + 4)^2}\)
Cancel common factors:
- Cancel \((x + 4)\) in the first fraction: \(\frac{x - 4}{2}\)
- Cancel \((x + 4)\) in the second fraction: \(\frac{\cancel{x + 4}}{\cancel{(x + 4)^2}}=\frac{1}{x + 4}\)
Multiply: \(\frac{x - 4}{2}\times\frac{1}{x + 4}=\frac{x - 4}{2(x + 4)}\), which is the target expression \(\frac{x - 4}{2(x + 4)}\) (wait, the target is \(\frac{x - 4}{2(x + 4)}\)? Wait, the given expression is \(\frac{x - 4}{2(x + 4)}\)? Wait, the original given expression is \(\frac{x - 4}{2(x + 4)}\)? Wait, the user - provided expression is \(\frac{x - 4}{2(x + 4)}\)? Wait, looking back: the given expression is \(\frac{x - 4}{2(x + 4)}\)? Wait, the problem says "the expression \(\frac{x - 4}{2(x + 4)}\)". And option C, when simplified, gives \(\frac{x - 4}{2(x + 4)}\). Wait, let's re - check the cancellation for option C:

\(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{x + 4}{(x + 4)^2}\)

- From the first fraction: \(\frac{(x - 4)\cancel{(x + 4)}}{2\cancel{(x + 4)}}=\frac{x - 4}{2}\)
- From the second fraction: \(\frac{\cancel{x + 4}}{\cancel{(x + 4)^2}}=\frac{1}{x + 4}\)
- Multiply: \(\frac{x - 4}{2}\times\frac{1}{x + 4}=\frac{x - 4}{2(x + 4)}\), which matches the given expression.

Step5: Simplify Option D

Option D: \(\frac{2x + 8}{x^2 - 16}\times\frac{x^2+8x + 16}{x + 4}\)
Substitute factored forms: \(\frac{2(x + 4)}{(x - 4)(x + 4)}\times\frac{(x + 4)^2}{x + 4}\)
Cancel common factors:
- Cancel \((x + 4)\) in the first fraction: \(\frac{2}{x - 4}\)
- Cancel \((x + 4)^2\) and \((x + 4)\) in the second fraction: \(x + 4\)
Multiply: \(\frac{2}{x - 4}\times(x + 4)=\frac{2(x + 4)}{x - 4}\), which is not the target.

Answer:

A. \(\frac{x^2 - 16}{2x + 8}\div\frac{x + 4}{x^2+8x + 16}\) (Wait, no, wait. Wait, in step 4, we saw that option C gives the target. Wait, I think I made a mistake in step 2. Let's re - do option A.

Option A: \(\frac{x^2 - 16}{2x + 8}\div\frac{x + 4}{x^2+8x + 16}=\frac{x^2 - 16}{2x + 8}\times\frac{x^2+8x + 16}{x + 4}=\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{(x + 4)^2}{x + 4}\)

Cancel \((x + 4)\) in \((x - 4)(x + 4)\) and \(2(x + 4)\): \(\frac{x - 4}{2}\)

Cancel \((x + 4)^2\) and \(x + 4\): \((x + 4)\)

Multiply: \(\frac{x - 4}{2}\times(x + 4)=\frac{(x - 4)(x + 4)}{2}=\frac{x^2 - 16}{2}\), which is not the target.

Option C: \(\frac{x^2 - 16}{2x + 8}\times\frac{x + 4}{x^2+8x + 16}=\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{x + 4}{(x + 4)^2}\)

Cancel \((x + 4)\) in the first fraction: \(\frac{x - 4}{2}\)

Cancel \((x + 4)\) in the second fraction: \(\frac{1}{x + 4}\)

Multiply: \(\frac{x - 4}{2(x + 4)}\), which is the target. So the correct option is C. Wait, but let's check again.

Wait, the given expression is \(\frac{x - 4}{2(x + 4)}\). Let's re - express option C:

\(\frac{x^2 - 16}{2x + 8}\times\frac{x + 4}{x^2+8x + 16}=\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{x + 4}{(x + 4)^2}\)

After canceling \((x + 4)\) from \((x - 4)(x + 4)\) and \(2(x + 4)\), we have \(\frac{x - 4}{2}\). Then, canceling \((x + 4)\) from \(x + 4\) and \((x + 4)^2\), we have \(\frac{1}{x + 4}\). Multiplying these two gives \(\frac{x - 4}{2(x + 4)}\), which matches the given expression. So the correct answer is C.

Wait, but in my first calculation of option A, I made a mistake. Let's confirm:

Option A: Division, so \(\frac{x^2 - 16}{2x + 8}\div\frac{x + 4}{x^2+8x + 16}=\frac{x^2 - 16}{2x + 8}\times\frac{x^2+8x + 16}{x + 4}\)

\(x^2 - 16=(x - 4)(x + 4)\), \(2x + 8 = 2(x + 4)\), \(x^2+8x + 16=(x + 4)^2\)

So substituting: \(\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{(x + 4)^2}{x + 4}\)

Cancel \((x + 4)\) in the first fraction: \(\frac{x - 4}{2}\)

Cancel \((x + 4)^2\) and \(x + 4\) in the second fraction: \((x + 4)\)

Multiply: \(\frac{x - 4}{2}\times(x + 4)=\frac{(x - 4)(x + 4)}{2}=\frac{x^2 - 16}{2}\), which is not the target.

Option C: Multiplication, \(\frac{x^2 - 16}{2x + 8}\times\frac{x + 4}{x^2+8x + 16}=\frac{(x - 4)(x + 4)}{2(x + 4)}\times\frac{x + 4}{(x + 4)^2}\)

Cancel \((x + 4)\) in the first fraction: \(\frac{x - 4}{2}\)

Cancel \((x + 4)\) in the second fraction: \(\frac{1}{x + 4}\)

Multiply: \(\frac{x - 4}{2(x + 4)}\), which is the target. So the correct option is C.