To write the given rational expression in its lowest terms, let's follow a step-by-step simplification process. The rational expression given is:
[tex]\[
\frac{q^2 - 49}{q^2 - 14q + 49}
\][/tex]
### Step 1: Factor the numerator and denominator
First, we factor both the numerator and the denominator.
#### Factor the numerator [tex]\( q^2 - 49 \)[/tex]
Notice that [tex]\( q^2 - 49 \)[/tex] is a difference of squares:
[tex]\[
q^2 - 49 = (q + 7)(q - 7)
\][/tex]
#### Factor the denominator [tex]\( q^2 - 14q + 49 \)[/tex]
Observe that [tex]\( q^2 - 14q + 49 \)[/tex] is a perfect square trinomial:
[tex]\[
q^2 - 14q + 49 = (q - 7)^2
\][/tex]
### Step 2: Rewrite the rational expression using the factors
Substitute the factored forms of the numerator and the denominator:
[tex]\[
\frac{q^2 - 49}{q^2 - 14q + 49} = \frac{(q + 7)(q - 7)}{(q - 7)^2}
\][/tex]
### Step 3: Simplify the rational expression
To simplify the rational expression, cancel the common factors in the numerator and the denominator. The common factor here is [tex]\( q - 7 \)[/tex]:
[tex]\[
\frac{(q + 7)(q - 7)}{(q - 7)(q - 7)} = \frac{q + 7}{q - 7}
\][/tex]
Thus, the simplified form of the given rational expression is:
[tex]\[
\frac{q + 7}{q - 7}
\][/tex]
In conclusion, the expression [tex]\( \frac{q^2 - 49}{q^2 - 14q + 49} \)[/tex] simplifies to [tex]\( \frac{q + 7}{q - 7} \)[/tex].
