# ISEE Upper Level Math : Variables and Exponents

## Example Questions

### Example Question #1 : How To Find The Exponent Of Variables

Simplify:

Possible Answers:

Correct answer:

Explanation:

Apply the power of a product property:

### Example Question #2 : How To Find The Exponent Of Variables

What is the coefficient of  in the expansion of .

Possible Answers:

Correct answer:

Explanation:

By the Binomial Theorem, if  is expanded, the coefficient of  is

.

Substitute : The coefficient of  is:

### Example Question #3 : How To Find The Exponent Of Variables

Simplify the expression:

Possible Answers:

Correct answer:

Explanation:

Apply the power of a power property twice:

### Example Question #2 : How To Find The Exponent Of Variables

What is the coefficient of  in the expansion of  ?

Possible Answers:

Correct answer:

Explanation:

By the Binomial Theorem, the  term of  is

,

making the coefficient of

.

We can set  in this expression:

### Example Question #5 : How To Find The Exponent Of Variables

What is the coefficient of  in the expansion of  ?

Possible Answers:

Correct answer:

Explanation:

By the Binomial Theorem, the  term of  is

.

Substitute  and this becomes

.

The coefficient is

.

### Example Question #6 : How To Find The Exponent Of Variables

Evaluate:

Possible Answers:

Correct answer:

Explanation:

We need to apply the power of power rule twice:

### Example Question #7 : How To Find The Exponent Of Variables

Solve for .

Possible Answers:

Correct answer:

Explanation:

Based on the power of a product rule we have:

The bases are the same, so we can write:

### Example Question #8 : How To Find The Exponent Of Variables

Simplify:

Possible Answers:

Correct answer:

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.

### Example Question #9 : How To Find The Exponent Of Variables

Simplify:

Possible Answers:

Correct answer:

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.

### Example Question #10 : How To Find The Exponent Of Variables

Simplify if and .

Possible Answers:

Correct answer:

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.