ISEE Upper Level Quantitative : Variables

Example Questions

Example Question #18 : Variables And Exponents

is a negative number. Which is the greater quantity?

(a)

(b)

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

(b) is the greater quantity

Explanation:

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 .

Explanation:

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 .

Explanation:

, 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

(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

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

Explanation:

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)

(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

(a) is the greater quantity

Explanation:

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

(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

(b) is the greater quantity

Explanation:

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)

(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

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

Explanation:

, 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:

The expression cannot be simplified further.

Explanation:

Group, then collect like terms.

Example Question #71 : Variables

Assume neither  nor  is zero.

Which is the greater quantity?

(a)

(b)

(a) and (b) are equal.

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Explanation:

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)

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.