ISEE Upper Level Quantitative : How to find the exponent of variables

Example Questions

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Simplify:

Explanation:

Example Question #2 : How To Find The Exponent Of Variables

Which is greater?

(a)

(b)

It is impossible to tell from the information given

(a) is greater

(b) is greater

(a) and (b) are equal

(b) is greater

Explanation:

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

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

The two quantities are equal.

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.

It is impossible to tell from the information given.

(a) and (b) are equal.

(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 #5 : How To Find The Exponent Of Variables

Which is the greater quantity?

(a)

(b)

It is impossble to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

(a) is greater.

Explanation:

Simplify the expression in (a):

Since

,

making (a) greater.

Example Question #62 : Variables

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

(a) is greater.

(b) is greater.

The two quantities are equal.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

Using the Binomial Theorem, if  is expanded, the  term is

.

This makes  the coefficient of .

We compare the values of this expression at  for both  and .

(a)  If  and , the coefficient is

.

This is the coefficient of .

(b) If  and , the coefficient is

.

This is the coefficient of .

(b) is the greater quantity.

Example Question #2 : How To Find The Exponent Of Variables

Consider the expression

Which is the greater quantity?

(a) The expression evaluated at

(b) The expression evaluated at

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Explanation:

Use the properties of powers to simplify the expression:

(a) If , then

(b) If , then

(b) is greater.

Example Question #8 : How To Find The Exponent Of Variables

Which of the following expressions is equivalent to

?

None of the other answers is correct.

None of the other answers is correct.

Explanation:

Use the square of a binomial pattern as follows:

This expression is not equivalent to any of the choices.

Example Question #9 : How To Find The Exponent Of Variables

Express   in terms of .

Explanation:

, so

, so

Example Question #63 : Variables

. Which is the greater quantity?

(a)

(b)

(a) is the greater quantity

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

(b) is the greater quantity

(a) and (b) are equal