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ISEE Upper Level Quantitative : Exponential Operations

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #1 : Exponential Operations

Simplify: $\left (y ^{3} \right )^{4}$

Possible Answers:

$y ^{12}$

$y ^{7}$

The correct answer is not among the other choices.

$y ^{64}$

$y ^{81}$

Correct answer:

$y ^{12}$

Explanation:

Apply the power of a power property:

$\left (y ^{3} \right )^{4} = y ^{3 \cdot 4} = y^{12}$

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Example Question #2 : Exponential Operations

Simplify the expression: $\left (5 a^{3}y ^{4} \right )^{2}$

Possible Answers:

$25a^{6} y^{8}$

$10a^{9} y^{16}$

$10a^{5} y^{6}$

$10a^{6} y^{8}$

$25a^{9} y^{16}$

Correct answer:

$25a^{6} y^{8}$

Explanation:

Apply the power of a product rule, then apply the power of a power rule:

$\left (5 a^{3}y ^{4} \right )^{2}$

$= 5^{2} \cdot \left (a^{3}\right )^{2} \cdot \left ( y ^{4} \right )^{2}$

$= 25 \cdot a^{ 3 \cdot 2 } \cdot y ^{ 4 \cdot 2 }$

$= 25 a^{ 6} y ^{ 8 }$

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Example Question #3 : Exponential Operations

Which of the following expressions is equal to $1,234^{0}$ ?

Possible Answers:

The expression is undefined.

1,234

$\frac{1}{1,234 }$

Correct answer:

Explanation:

Any nonzero number raised to the power of 0 is equal to 1.

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Example Question #4 : Exponential Operations

Which quantity is greater?

(a) $(-5) ^{3}$

(b) $(-5) ^{-3}$

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(a) is greater.

Correct answer:

(b) is greater.

Explanation:

(a) $(-5) ^{3} = (-5) \cdot (-5) \cdot (-5) = 25 \cdot (-5) = -125$

(b) $(-5) ^{-3} = \frac{1}{(-5) ^{3}} = \frac{1}{-125} =- \frac{1}{125}$

(b) is the greater quantity.

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Example Question #5 : Exponential Operations

is positive.

Which is the greater quantity?

(a) $100 ^{2x}$

(b) $1,000 ^{x}$

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

Correct answer:

(a) is greater

Explanation:

Use the power of a power property:

(a) $100 ^{2x} = \left (10^{2} \right ) ^{2x} = 10^{2\cdot 2x }= 10^{4x }$

(b) $1,000 ^{x} =\left ( 1 0 ^{3} \right ) ^{x} = 1 0 ^{3x}$

Since x > 0 , 4x > 3x . Subsequently,

$100 ^{2x} = 10^{4x } > 10^{3x } = 1,000^{x }$ ,

making (a) greater

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Example Question #6 : How To Multiply Exponents

Which is the greater quantity?

(a) $2 ^{100}$

(b) $4^{50}$

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

$4^{50} = \left ( 2 ^{2} \right )^{50} = 2 ^{ 2\; \cdot \;50 } = 2 ^{ 100 }$

The two quantities are equal.

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Example Question #7 : 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:

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

The quantities in both columns are equal.

The quantity in Column A is greater.

The quantity in Column B is greater.

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 #9 : How To Multiply Exponents

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 #10 : How To Multiply Exponents

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

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

(B) is greater

(A) and (B) are equal

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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ISEE Upper Level Quantitative : Exponential Operations

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Exponential Operations

Example Question #2 : Exponential Operations

Example Question #3 : Exponential Operations

Example Question #4 : Exponential Operations

Example Question #5 : Exponential Operations

Example Question #6 : How To Multiply Exponents

Example Question #7 : How To Multiply Exponents

Example Question #8 : How To Multiply Exponents

Example Question #9 : How To Multiply Exponents

Example Question #10 : How To Multiply Exponents

All ISEE Upper Level Quantitative Resources

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