How to find the exponent of variables

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ISEE Upper Level Quantitative Reasoning › How to find the exponent of variables

Questions 1 - 10
1

Simplify if and .

Explanation

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.

2

Simplify the expression:

Explanation

Apply the power of a power property twice:

3

Simplify:

Explanation

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.

4

Evaluate .

Explanation

By the Power of a Product Principle,

Also, by the Power of a Power Principle,

Combining these ideas, then substituting:

5

Solve for .

Explanation

Based on the power of a product rule we have:

The bases are the same, so we can write:

6

Evaluate .

Explanation

By the Power of a Power Principle,

So

Also, by the Power of a Product Principle,

, so, substituting,

.

7

Simplify:

Explanation

Apply the power of a product property:

8

What is the coefficient of in the expansion of ?

Explanation

By the Binomial Theorem, the term of is

,

making the coefficient of

.

We can set in this expression:

9

Evaluate .

Explanation

10

What is the coefficient of in the expansion of ?

Explanation

By the Binomial Theorem, the term of is

.

Substitute and this becomes

.

The coefficient is

.

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