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ISEE Upper Level Quantitative : Numbers and Operations

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : How To Multiply Exponents

Two quantities are given - one in Column A and the other in Column B. Compare the quantities in the two columns.

Assume, in both columns, that $x\neq0$ .

Column A Column B

$x^{4}\cdot x^2$ $\frac{x^{11}}{x^{2}}$

Possible Answers:

The quantity in Column B is greater.

The quantities in both columns are equal.

The quantity in Column A is greater.

There is no way to determine the relationship between the columns.

Correct answer:

There is no way to determine the relationship between the columns.

Explanation:

Column A gives simplifies to give us $\ x^{6}$ , and Column B simplifies to give us $x^{9}$ . At first glance, Column B is greater, as it would be for all answers greater than 1. However, if x=1 , the two columns are equal. Furthermore, if is negative, or a fraction, Column A is greater. Thus, since we could arrive at all three answers by using different numbers, we cannot determine the answer conclusively.

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Example Question #8 : How To Multiply Exponents

Which of the following expressions is equivalent to

$\left (\sqrt{1,001}-\sqrt{999} \right ) \left ( \sqrt{1,001}+\sqrt{999} \right )$ ?

Possible Answers:

$\sqrt{2}$

$20\sqrt{5}$

2,000

$2,000 - 3 \sqrt{111,111}$

Correct answer:

Explanation:

Use the difference of squares pattern as follows:

$\left (\sqrt{1,001}-\sqrt{999} \right ) \left ( \sqrt{1,001}+\sqrt{999} \right )$

$= \left (\sqrt{1,001} \right ) ^{2} - \left (\ \sqrt{999} \right ) ^{2}$

=1,001 - 999 = 2

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Example Question #5 : Exponential Operations

Column A Column B

$\left ( \frac{5x^2}{25y^4} \right )^{0}$

Possible Answers:

The quantity in Column A is greater.

The quantities in both columns are equal.

No relationship between the columns can be determined.

The quantity in Column B is greater.

Correct answer:

The quantity in Column A is greater.

Explanation:

Anything raised to zero is equal to 1. Therefore, Column A has to be greater because 1 is greater than 0.

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Example Question #111 : Numbers And Operations

44,000,000 can be written in scientific notation as $a \times 10 ^{N}$ for some a, N .

Which is the greater quantity?

(A)

(B) 8

Possible Answers:

(A) is greater

(B) is greater

(A) and (B) are equal

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

Correct answer:

(B) is greater

Explanation:

To write 44,000,000 in scientifc notation, write the implied decimal point after the final "0", then move it left until it is after the first nonzero digit (the first "4").

$44 000 000. \Rightarrow 4.4 000 000$

This requires a displacement of seven places, so

$44,000,000 = 4.4 \times 10^{7}$

N = 7 < 8 , and (B) is greater.

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Example Question #11 : Exponential Operations

Give the result:

$333,335 ^{2} - 333,331 ^{2}$

Possible Answers:

2,000,004

4,444,444

4,000,004

2,469,754

2,666,664

Correct answer:

2,666,664

Explanation:

For any two numbers a ,b ,

$a^{2} - b ^{2} = (a+b) (a - b)$

We can solve this problem most easily by taking advantage of this pattern, setting

a = 333,335 , b =333,331 :

$333,335 ^{2} - 333,331 ^{2}$

$= \left (333,335 + 333,331 \right )\left (333,335 - 333,331 \right )$

$=666,666 \times 4$

=2,666,664

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Example Question #11 : Exponents

Raise $8 \times 10 ^{-3}$ to the fourth power and give the result in scientific notation.

Possible Answers:

$4.096 \times 10^{-84}$

$4.096 \times 10^{-9}$

$3.2 \times 10^{-11}$

$4.096 \times 10^{-15}$

$3.2 \times 10^{-13}$

Correct answer:

$4.096 \times 10^{-9}$

Explanation:

Use the properties of exponents to raise the number to the fourth power:

$\left (8 \times 10 ^{-3} \right ) ^{4}$

$= 8 ^{4} \times \left ( 10 ^{-3} \right ) ^{4}$

$= 4,096 \times 10 ^{-3 \times 4 }$

$= 4,096 \times 10 ^{-12 }$

This is not in scientific notation, so adjust:

$4.096 \times 10^{3} \times 10 ^{-12 }$

$= 4.096 \times 10^{3 + (-12)}$

$= 4.096 \times 10^{-9}$

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Example Question #13 : Exponential Operations

Give the result:

$1,000,001 ^{2} - 999,999^{2}$

Possible Answers:

8,000,000

2,000,000

6,000,000

1,000,000

4,000,000

Correct answer:

4,000,000

Explanation:

For any two numbers a ,b ,

$a^{2} - b ^{2} = (a+b) (a - b)$

We can solve this problem most easily by taking advantage of this pattern, setting

a = 1,000,001 , b = 999,999 :

$1,000,001 ^{2} - 999,999^{2}$

$= \left ( 1,000,001+ 999,999 \right )\left ( 1,000,001- 999,999 \right )$

$= 2,000,000 \times 2 = 4,000,000$

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Example Question #11 : Exponential Operations

Give the cube of $6 \times 10 ^{4}$ in scientific notation.

Possible Answers:

$2.16 \times 10 ^{13}$

$2.16 \times 10 ^{12 }$

$2.16 \times 10 ^{14 }$

$1.8 \times 10 ^{12 }$

$1.8 \times 10 ^{13 }$

Correct answer:

$2.16 \times 10 ^{14 }$

Explanation:

$\left (6 \times 10 ^{4} \right ) ^{3}$

$= 6^{3} \times \left (10 ^{4} \right ) ^{3}$

$=216 \times 10 ^{4 \times 3 }$

$=216 \times 10 ^{12 }$

This is not in scientific notation, so adjust:

$2.16 \times 10 ^{2 } \times 10 ^{12 }$

$=2.16 \times 10 ^{2 + 12 }$

$=2.16 \times 10 ^{14 }$

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Example Question #11 : Exponents

Which expression is equal to 65,000?

Possible Answers:

$6.5*10^{4}$

$6.5*10^{5}$

$6.5*10^{3}$

$65*10^{4}$

Correct answer:

$6.5*10^{4}$

Explanation:

$6.5*10^{4}$ is equal to 6.5*10,000

Move the decimal one place to the right for each number of the exponent with a base ten.

For example, $6.5 \ast10^{^{1}}=65$ , $6.5*10^{^{2}}=650$ , etc.

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Example Question #11 : Exponential Operations

$x^{3} = - 7$

$y ^{6} = 13$

Evaluate $(xy) ^{6}$ .

Possible Answers:

637

Correct answer:

637

Explanation:

By the Power of a Product Principle,

$(xy) ^{6} = x ^{6} \cdot y ^{6}$

Also, by the Power of a Power Principle,

$(x^{3} ) ^{2} = x ^{3 \cdot 2 } = x ^{6}$

Combining these ideas,

$(xy) ^{6} = (x^{3} ) ^{2} \cdot y ^{6} = (-7 ) ^{2} \cdot 13 = 7 ^{2} \cdot 13 = 49 \cdot 13 = 637$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Numbers and Operations

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Multiply Exponents

Example Question #8 : How To Multiply Exponents

Example Question #5 : Exponential Operations

Example Question #111 : Numbers And Operations

Example Question #11 : Exponential Operations

Example Question #11 : Exponents

Example Question #13 : Exponential Operations

Example Question #11 : Exponential Operations

Example Question #11 : Exponents

Example Question #11 : Exponential Operations

All ISEE Upper Level Quantitative Resources

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