Simplify each and state the excluded values.

Ask by Stanley Hodgson. in the United States

Dec 07,2024

Question

Simplify each and state the excluded values. $$ \frac{7 n^{2}+14 n}{n+8}: \frac{n+8}{n^{2}+10 n+16} $$

UpStudy AI · Accepted Answer

To simplify the expression

$$
\frac{7 n^{2}+14 n}{n+8} : \frac{n+8}{n^{2}+10 n+16},
$$

we can rewrite the division as multiplication by the reciprocal:

$$
\frac{7 n^{2}+14 n}{n+8} \cdot \frac{n^{2}+10 n+16}{n+8}.
$$

Next, we will simplify each part of the expression.

1. **Factor the numerator $$7n^2 + 14n$$**:

$$
7n^2 + 14n = 7n(n + 2).
$$

2. **Factor the denominator $$n^2 + 10n + 16$$**:

To factor $$n^2 + 10n + 16$$, we look for two numbers that multiply to $$16$$ and add to $$10$$. These numbers are $$2$$ and $$8$$:

$$
n^2 + 10n + 16 = (n + 2)(n + 8).
$$

Now we can substitute these factorizations back into the expression:

$$
\frac{7n(n + 2)}{n + 8} \cdot \frac{(n + 2)(n + 8)}{n + 8}.
$$

3. **Cancel common factors**:

The $$n + 8$$ in the numerator and denominator cancels out, as does the $$n + 2$$:

$$
= 7n \cdot (n + 2).
$$

Thus, the simplified expression is:

$$
7n.
$$

4. **State the excluded values**:

The excluded values come from the original denominators. We need to find values of $$n$$ that make the denominators zero:

- From $$n + 8$$, we have $$n + 8 = 0 \Rightarrow n = -8$$.
- From $$n^2 + 10n + 16$$, we set it to zero:

$$
n^2 + 10n + 16 = 0.
$$

Using the quadratic formula $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$:

$$
n = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 16}}{2 \cdot 1} = \frac{-10 \pm \sqrt{100 - 64}}{2} = \frac{-10 \pm \sqrt{36}}{2} = \frac{-10 \pm 6}{2}.
$$

This gives us:

$$
n = \frac{-4}{2} = -2 \quad \text{and} \quad n = \frac{-16}{2} = -8.
$$

Thus, the excluded values are $$n = -8$$ and $$n = -2$$.

In conclusion, the simplified expression is:

$$
7n,
$$

with excluded values:

$$
n = -8, \quad n = -2.
$$
The simplified expression is $$7n$$, and the excluded values are $$n = -8$$ and $$n = -2$$.

Simplify each and state the excluded values.

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions