Video Transcript
Simplify 𝑎 to the power of 𝑥 plus
nine plus 𝑎 to the power of 𝑥 plus eight plus 𝑎 to the power of 𝑥 plus seven
over 𝑎 to the power of 𝑥 minus two plus 𝑎 to the power of 𝑥 minus three plus 𝑎
to the power of 𝑥 minus four.
In order to simplify this rational
expression, we’ll need to recall exponent properties and some factoring
principles.
Since each term in the expression
is of the form 𝑎 to the power of 𝑚 plus 𝑛, we can start by applying the product
rule for exponents. The product rule states that 𝑎 to
the power of 𝑚 plus 𝑛 is equal to 𝑎 to the power of 𝑚 multiplied by 𝑎 to the
power of 𝑛, where 𝑎 is a nonzero real number and 𝑚 and 𝑛 are integers. Applying this rule to each term
results in the following expression. 𝑎 to the power of 𝑥 times 𝑎 to
the power of nine plus 𝑎 to the power of 𝑥 times 𝑎 to the power of eight plus 𝑎
to the power of 𝑥 times 𝑎 to the power of seven over 𝑎 to the power of 𝑥 times
𝑎 to the power of negative two plus 𝑎 to the power of 𝑥 times 𝑎 to the power of
negative three plus 𝑎 to the power of 𝑥 times 𝑎 to the power of negative
four.
Next, we can pull the common factor
of 𝑎 to the power of 𝑥 out to the front in both the numerator and the
denominator. The result of this factoring is the
expression 𝑎 to the power of 𝑥 times 𝑎 to the power of nine plus 𝑎 to the power
of eight plus 𝑎 to the power of seven over 𝑎 to the power of 𝑥 times 𝑎 to the
power of negative two plus 𝑎 to the power of negative three plus 𝑎 to the power of
negative four. Then, we can cancel the common
factor of 𝑎 to the power of 𝑥 from the numerator and denominator.
What remains does not have any
obvious common factors. However, after examining the
expression, we notice a pattern. The exponents decrease by one in
each consecutive term from left to right. Here we can use a bit of creative
factoring to reveal the common structure behind the three terms in the numerator and
denominator. We will factor 𝑎 to the power of
seven from all the three terms in the numerator and factor 𝑎 to the power of
negative four from the denominator.
To properly factor 𝑎 to the power
of seven from each term in the numerator, we will use the quotient rule for
exponents, which states that 𝑎 to the power of 𝑚 over 𝑎 to the power of 𝑛 is
equal to 𝑎 to the power of 𝑚 minus 𝑛, for any nonzero real number 𝑎 and integers
𝑚 and 𝑛. To factor out 𝑎 to the power of
seven from 𝑎 to the power of nine, we are essentially dividing 𝑎 to the power of
nine by 𝑎 to the power of seven. According to the quotient rule,
this simplifies to 𝑎 squared.
Next, we’ll need to find the
quotient of 𝑎 to the power of eight and 𝑎 to the power of seven, which simplifies
to 𝑎 to the power of one, or just 𝑎. Then, we need to find the quotient
of 𝑎 to the power of seven and 𝑎 to the power of seven, which simplifies to one,
since the numerator and denominator are the same.
Now we will move on to the
denominator, dividing each term by 𝑎 to the power of negative four. First, we take 𝑎 to the power of
negative two divided by 𝑎 to the power of negative four. Using the quotient property, we can
write this expression as 𝑎 to the power of negative two minus negative four, which
simplifies to 𝑎 squared. Then, we take 𝑎 to the power of
negative three divided by 𝑎 to the power of negative four, which simplifies to
𝑎. And finally, we take the quotient
of 𝑎 to the power of negative four divided by 𝑎 to the power of negative four,
which simplifies to one.
As we now see, the numerator and
denominator share a common trinomial factor of 𝑎 squared plus 𝑎 plus one. After canceling this common factor,
we are left with 𝑎 to the power of seven over 𝑎 to the power of negative four.
To finish simplifying, we once
again use the quotient rule and find the exponent of 𝑎 by taking seven minus
negative four. The result is 𝑎 to the power of
11. This is the full simplification of
𝑎 to the power of 𝑥 plus nine plus 𝑎 to the power of 𝑥 plus eight plus 𝑎 to the
power of 𝑥 plus seven over 𝑎 to the power of 𝑥 minus two plus 𝑎 to the power of
𝑥 minus three plus 𝑎 to the power of 𝑥 minus four.
