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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 #13 : Exponential Operations

$\frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}}$

Possible Answers:

The expression is undefined.

Correct answer:

The expression is undefined.

Explanation:

$\frac{ \left ( 30 -2 \cdot 15\right )^{0}}{\left ( 30 -15\right )^{0}} = \frac{ \left ( 30 -30\right )^{0}}{15^{0}}=\frac{ 0^{0}}{15^{0}}$

The numerator is undefined, since 0 raised to the power of 0 is an undefined quantity. Therefore, the entire expression is undefined.

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Example Question #2 : How To Divide Exponents

Column A Column B

$\left(\frac{1}{16}\right)^\frac{1}{2}$ $\left(\frac{1}{16}\right)^{-\frac{1}{2}}$

Possible Answers:

The quantity in Column B is greater.

There is not enough information to determine the relationship between the quantities.

The quantity in Column A is greater.

The quantities are equal.

Correct answer:

The quantity in Column B is greater.

Explanation:

Let's simplify both quantities first before we compare them. $\left(\frac{1}{16}\right)^\frac{1}{2}$ becomes $\sqrt{\frac{1}{16}}$ because the fractional exponent indicates a square root. We can simplify that by knowing that we can take the square roots of both the numerator and denominator, as shown by: $\frac{\sqrt{1}}{\sqrt{16}}$ . We can simplify further by taking the square roots (they're perfect squares) and get $\frac{1}{4}$ . Then, let's simplify Column B. To get rid of the negative exponent, we put the numerical expression on the denominator. There's still the fractional exponent at play, so we'll have a square root as well. It looks like this now: $\frac{1}{\sqrt{\frac{1}{16}}}$ . We already simplified $\sqrt{\frac{1}{16}}$ , so we can just plug in our answer, $\frac{1}{4}$ , into the denominator. Since we don't want a fraction in the denominator, we can multiply by the reciprocal of $\frac{1}{4}$ , which is 4 to get 1(4) , which is just 4. Therefore, Column B is greater.

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Example Question #1 : How To Divide Exponents

Give the reciprocal of $1.6 \times 10 ^{-7}$ in scientific notation.

Possible Answers:

$6.25 \times 10^{-6 }$

$6.25 \times 10^{-7 }$

$6.25 \times 10^{8 }$

$6.25 \times 10^{6 }$

$6.25 \times 10^{7 }$

Correct answer:

$6.25 \times 10^{6 }$

Explanation:

The reciprocal of $1.6 \times 10 ^{-7}$ is the quotient of 1 and the number;

$\frac{1}{ 1.6 \times 10 ^{-7} }$

$= \frac{1 \times 10^{0}}{ 1.6 \times 10 ^{-7} }$

$= \frac{1 }{ 1.6 } \times \frac{ 10^{0}}{ 10 ^{-7} }$

$=0.625 \times 10^{0- (-7) }$

$=0.625 \times 10^{7 }$

This is not in scientific notation, so adjust.

$6.25 \times 10^{-1 } \times 10^{7 }$

$= 6.25 \times 10^{-1+7 }$

$= 6.25 \times 10^{6 }$

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Example Question #1 : How To Divide Exponents

Give the reciprocal of $3.2 \times 10 ^{9}$ in scientific notation.

Possible Answers:

$3.125 \times 10 ^{-9}$

$-3.2 \times 10 ^{-9}$

$-3.2 \times 10 ^{-10}$

$3.125 \times 10 ^{-10}$

$-3.125 \times 10 ^{9}$

Correct answer:

$3.125 \times 10 ^{-10}$

Explanation:

The reciprocal of $3.2 \times 10 ^{9}$ is the quotient of 1 and the number, or

$\frac{1}{ 3.2 \times 10 ^{9} }$

$= \frac{1 \times 10^{0}}{ 3.2 \times 10 ^{9}}$

$= \frac{1 }{ 3.2 } \times \frac{ 10^{0}}{ 10 ^{9} }$

$=0.3125 \times 10 ^{-9}$

This is not in scientific notation, so adjust:

$3.125 \times 10 ^{-1}\times 10 ^{-9}$

$=3.125 \times 10 ^{-1+ (-9)}$

$=3.125 \times 10 ^{-10}$

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Example Question #122 : Numbers And Operations

x> 1

Column A Column B

$\frac{x^3}{x^2}$ $\sqrt{x}$

Possible Answers:

The relationship between the columns cannot be determined.

The quantities in both columns are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

Correct answer:

The quantity in Column A is greater.

Explanation:

You can simplify Column A first. When you're dividing with exponents and bases are the same, subtract the exponents. Therefore, it simplifies to x. We know that x is positive since it is greater than 1. X is greater than $\sqrt{x}$ . Try plugging in a number to test. 25 is greater than $\sqrt{25}$ , which is 5. Even 1.1 is greater than $\sqrt{1.1}$ . Therefore, Column A is greater.

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Example Question #1 : How To Add 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

2x^2+6x^2 11x^2-x^2

Possible Answers:

The relationship cannot be determined from the info given.

The quantities in both columns are equal.

The quantity in Column B is greater.

The quantity in Column A is greater.

Correct answer:

The quantity in Column B is greater.

Explanation:

When you are adding and subtracting terms with exponents, you combine like terms. Since both columns have expressions with the same exponent throughout, you are good to just look at the coefficients. Remember, a coefficient is the number in front of a variable. Therefore, Column A is 8x^2 since 2+6=8 . Column B is 10x^2 since 11-1=10 . We can see that Column B is greater.

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Example Question #121 : Numbers And Operations

Add all of the perfect squares between 50 and 100 inclusive.

Possible Answers:

194

294

The correct answer is not among the other choices.

245

145

Correct answer:

245

Explanation:

The perfect squares between 50 and 100 inclusive are

$8^{2} = 64$

$9^{2} = 81$

$10 ^{2} = 100$

Their sum is 64 + 81 + 100 = 245

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Example Question #21 : Exponents

Which is the greater quantity?

(A) The sum of the first ten perfect square integers

(B) The sum of the first five perfect cube integers

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:

(A) is greater

Explanation:

The sum of the first ten perfect square integers:

$\begin{matrix} \;\; \; 1\\ \;\; \; 4\\ \;\; \; 9\\ \;16\\ \;25\\ \;36\\ \;49\\ \;64\\ \;81\\ \underline{100}\\ 385 \end{matrix}$

The sum of the first five perfect cube integers:

$\begin{matrix} \;\; \; 1\\ \;\; \; 8\\ \;27\\ \;64\\ \underline{125}\\ 225 \end{matrix}$

(A) is greater.

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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 #13 : Exponential Operations

Example Question #2 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

Example Question #122 : Numbers And Operations

Example Question #1 : How To Add Exponents

Example Question #121 : Numbers And Operations

Example Question #21 : Exponents

All ISEE Upper Level Quantitative Resources

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