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ISEE Upper Level Quantitative : How to divide exponents

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 #12 : 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 #2 : How To Divide Exponents

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

Possible Answers:

$6.25 \times 10^{6 }$

$6.25 \times 10^{7 }$

$6.25 \times 10^{-7 }$

$6.25 \times 10^{8 }$

$6.25 \times 10^{-6 }$

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 #2 : How To Divide Exponents

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

Possible Answers:

$-3.125 \times 10 ^{9}$

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

$3.125 \times 10 ^{-9}$

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

$3.125 \times 10 ^{-10}$

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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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to divide exponents

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #12 : Exponential Operations

Example Question #2 : How To Divide Exponents

Example Question #2 : How To Divide Exponents

Example Question #2 : How To Divide Exponents

Example Question #122 : Numbers And Operations

All ISEE Upper Level Quantitative Resources

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