# ISEE Upper Level Quantitative : Variables and Exponents

## Example Questions

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### Example Question #1 : How To Add Exponential Variables

Simplify:     The expression cannot be simplified further Explanation:

Group and combine like terms :    ### Example Question #1 : Variables And Exponents Which is the greater quantity?

(a) (b) (b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

Explanation: Since and have different signs, , and, subsequently, Therefore, This makes (b) the greater quantity.

### Example Question #1 : How To Add Exponential Variables

Assume that and are not both zero. Which is the greater quantity?

(a) (b) (a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

It is impossible to tell from the information given.

Explanation:

Simplify the expression in (a):    Therefore, whether (a) or (b) is greater depends on the values of and , neither of which are known.

### Example Question #4 : How To Add Exponential Variables Which is the greater quantity?

(a) (b) (b) is greater

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

It is impossible to tell from the information given

Explanation:

We give at least one positive value of for which (a) is greater and at least one positive value of for which (b) is greater.

Case 1: (a) (b) Case 2: (a) (b) Therefore, either (a) or (b) can be greater.

### Example Question #5 : How To Add Exponential Variables

Assume all variables to be nonzero.

Simplify:    None of the answer choices are correct. None of the answer choices are correct.

Explanation:

Any nonzero expression raised to the power of 0 is equal to 1. Therefore, .

None of the given expressions are correct.

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

Simplify:       Explanation: ### Example Question #1 : How To Find The Exponent Of Variables Which is greater?

(a) (b) (a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

(b) is greater

Explanation:

If , then and  , so by transitivity, , and (b) is greater

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

Expand: Which is the greater quantity?

(a) The coefficient of (b) The coefficient of It is impossible to tell from the information given.

(a) is greater.

The two quantities are equal.

(b) is greater.

The two quantities are equal.

Explanation:

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

(a) Substitute : The coerfficient of is .

(b) Substitute : The coerfficient of is .

The two are equal.

### Example Question #61 : Variables

Which is greater?

(a) (b) (b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Explanation: A negative number to an odd power is negative, so the expression in (a) is negative. The expression in (b) is positive since the base is positive. (b) is greater.

### Example Question #1 : Variables And Exponents Which is the greater quantity?

(a) (b) (a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossble to tell from the information given.

(a) is greater.

Explanation:

Simplify the expression in (a):     Since  ,

making (a) greater.

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