### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #212 : Algebraic Concepts

Simplify:

**Possible Answers:**

The expression cannot be simplified further

**Correct answer:**

Group and combine like terms :

### Example Question #2 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

(a) is greater.

(a) and (b) are equal.

**Correct answer:**

(b) is greater.

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)

**Possible Answers:**

(a) is greater.

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

**Correct answer:**

It is impossible to tell from the information given.

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 #51 : Variables

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

**Correct answer:**

It is impossible to tell from the information given

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 : Variables And Exponents

Assume all variables to be nonzero.

Simplify:

**Possible Answers:**

None of the answer choices are correct.

**Correct answer:**

None of the answer choices are correct.

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:

**Possible Answers:**

**Correct answer:**

### Example Question #1 : Variables And Exponents

Which is greater?

(a)

(b)

**Possible Answers:**

(b) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

**Correct answer:**

(b) is greater

If , then and

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

### Example Question #1 : Variables And Exponents

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

**Possible Answers:**

(b) is greater.

It is impossible to tell from the information given.

The two quantities are equal.

(a) is greater.

**Correct answer:**

The two quantities are equal.

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)

**Possible Answers:**

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(b) is greater.

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)

**Possible Answers:**

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossble to tell from the information given.

**Correct answer:**

(a) is greater.

Simplify the expression in (a):

Since ,

,

making (a) greater.

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