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ISEE Upper Level Quantitative : 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 #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 : How To Multiply Exponents

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

Possible Answers:

$2.16 \times 10 ^{12 }$

$2.16 \times 10 ^{14 }$

$1.8 \times 10 ^{12 }$

$2.16 \times 10 ^{13}$

$1.8 \times 10 ^{13 }$

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:

$65*10^{4}$

$6.5*10^{4}$

$6.5*10^{5}$

$6.5*10^{3}$

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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Example Question #1 : How To Divide Exponents

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

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 #1 : How To Divide Exponents

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

Possible Answers:

$3.125 \times 10 ^{-10}$

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

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

$-3.125 \times 10 ^{9}$

$3.125 \times 10 ^{-9}$

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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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Exponents

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 : Exponential Operations

Example Question #11 : Exponents

Example Question #13 : Exponential Operations

Example Question #11 : How To Multiply Exponents

Example Question #11 : Exponents

Example Question #11 : Exponential Operations

Example Question #1 : How To Divide Exponents

Example Question #2 : How To Divide Exponents

Example Question #2 : How To Divide Exponents

Example Question #1 : How To Divide Exponents

All ISEE Upper Level Quantitative Resources

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