ISEE Upper Level Quantitative Reasoning › How to find the exponent of variables
Simplify if and
.
Begin by factoring the numerator and denominator. can be factored out of each term.
can be canceled, since it appears in both the numerator and denomintor.
Next, factor the numerator.
Simplify.
Simplify the expression:
Apply the power of a power property twice:
Simplify:
First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.
Apply the exponent within the parentheses and simplify.
This fraction cannot be simplified further.
Evaluate .
By the Power of a Product Principle,
Also, by the Power of a Power Principle,
Combining these ideas, then substituting:
Solve for .
Based on the power of a product rule we have:
The bases are the same, so we can write:
Evaluate .
By the Power of a Power Principle,
So
Also, by the Power of a Product Principle,
, so, substituting,
.
Simplify:
Apply the power of a product property:
What is the coefficient of in the expansion of
?
By the Binomial Theorem, the term of
is
,
making the coefficient of
.
We can set in this expression:
Evaluate .
What is the coefficient of in the expansion of
?
By the Binomial Theorem, the term of
is
.
Substitute and this becomes
.
The coefficient is
.