### All ISEE Upper Level Quantitative Resources

## Example Questions

### Example Question #18 : Variables And Exponents

is a negative number. Which is the greater quantity?

(a)

(b)

**Possible Answers:**

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

(b) is the greater quantity

Any nonzero number raised to an even power, such as 4, is a positive number. Therefore,

is the product of a negative number and a positive number, and is therefore negative.

By the same reasoning, is a positive number.

It follows that .

### Example Question #19 : Variables And Exponents

Evaluate .

**Possible Answers:**

**Correct answer:**

By the Power of a Power Principle,

By way of the Power of a Quotient Principle,

.

### Example Question #20 : Variables And Exponents

and are both real numbers.

Evaluate .

**Possible Answers:**

**Correct answer:**

, as the product of a sum and a difference, can be rewritten using the difference of squares pattern:

By the Power of a Power Principle,

Therefore, is a square root of - that is, a square root of 121. 121 has two square roots, and 121, but since is real, must be the positive choice, 11. Similarly, is the positive square root of 81, which is 9.

The above expression can be evaluated as

.

### Example Question #21 : Variables And Exponents

Which is the greater quantity?

(a)

(b) 37

**Possible Answers:**

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

It is impossible to determine which is greater from the information given

Multiply the polynomials through distribution:

The absolute value of is 4, so either or . Likewise, or .

If and , we see that

If and , we see that

In the first scenario, ; in the second, . This makes the information insufficient.

### Example Question #22 : Variables And Exponents

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

(a) is the greater quantity

It is impossible to determine which is greater from the information given

(b) is the greater quantity

**Correct answer:**

(a) is the greater quantity

The absolute value of is 4, so either or . Likewise, or . However, since and , it follows that regardless, and .

As the product of the sum and the difference of the same two expressions, can be rewritten as the difference of the squares of the expressions:

Using the Power of a Product Principle:

Substituting,

Similarly,

Therefore, .

### Example Question #23 : Variables And Exponents

Which is the greater quantity?

(a)

(b) 16

**Possible Answers:**

(a) and (b) are equal

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

**Correct answer:**

(b) is the greater quantity

Multiply the polynomials through distribution:

Collecting like terms, the above becomes

By the Power of a Power Principle,

This makes a square root (positive or negative) of , or 81, so

or

We can not eliminate either since an odd power of a number can have any sign, and we are not given the sign of .

By similar reasoning, either

or

can assume one of four values, depending on which values of and are selected:

Regardless of the choice of and , .

### Example Question #71 : Variables

Define .

is a function with the set of all real numbers as its domain.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(b) is the greater quantity

**Correct answer:**

It is impossible to determine which is greater from the information given

, so .

By definition,

.

Since and , we can determine that

.

However, this does not tell us the value of at . Therefore, we do not know whether or , if either, is the greater.

### Example Question #72 : Variables

Simplify the expression:

**Possible Answers:**

The expression cannot be simplified further.

**Correct answer:**

Group, then collect like terms.

### Example Question #71 : Variables

Assume neither nor is zero.

Which is the greater quantity?

(a)

(b)

**Possible Answers:**

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

**Correct answer:**

(a) and (b) are equal.

Simplify the expression in (a):

regardless of the values of the variables, and (a) and (b) are equal.

### Example Question #74 : Variables

is a positive number. is a negative number.

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:**

(b) is greater.

is a positive number and is a negative number, so and . Therefore,

.

(b) is always greater.