# 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 #2 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

Since  and  have different signs,

, and, subsequently,

Therefore,

This makes (b) the greater quantity.

### Example Question #3 : Variables And Exponents

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

(a)

(b)

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) 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 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

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 #1 : 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.

Simplify:

Explanation:

### Example Question #7 : Variables And Exponents

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

The two quantities are equal.

(a) is greater.

It is impossible to tell from the information given.

(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.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(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 #2 : How To Find The Exponent Of Variables

Which is the greater quantity?

(a)

(b)

(b) is greater.

It is impossble to tell from the information given.

(a) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

Simplify the expression in (a):

Since

,

making (a) greater.

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