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ISEE Upper Level Math : Exponents

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

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All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Exponents

Simplify:

3a^3-a^2+4a^2-8a^3

Possible Answers:

a^2(-5a+3)

a(-5a+3)

a^2(5a-3)

a^2(5a+3)

a^2(-5a+2)

Correct answer:

a^2(-5a+3)

Explanation:

In order to add or subtract exponential terms, both the base and the exponent must be the same. So we can write:

3a^3-a^2+4a^2-8a^3=(3a^3-8a^3)+(4a^2-a^2)

=-5a^3+3a^2=a^2(-5a+3)

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Example Question #2 : Exponents

Evaluate:

5^2+5^3-7^2-3(7^2)-4(5^3)

Possible Answers:

546

550

Correct answer:

Explanation:

In order to add or subtract exponential terms, both the base and the exponent must be the same. So we can write:

5^2+5^3-7^2-3(7^2)-4(5^3)=5^2+(5^3-4(5^3))-(7^2+3(7^2))

=5^2+-3(5^3)-4(7^2)

Now they must be multiplied out before they can be added:

$=5\times 5- 3\times 5\times 5\times 5-4\times 7\times 7$

$=25-3\times 125-4\times 49$

=25-375-196=-546

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Example Question #1 : Exponential Operations

Subtract and simplify:

$\left ( 11 x^{2}- 8x-7 \right )- \left (-7 x^{2}-5x+19\right )$

Possible Answers:

$4x^{2}-3x+12$

$18x^{2}-3x+12$

$18x^{2}-3x-26$

$18x^{2}-13x-26$

$4x^{2}-3x-26$

Correct answer:

$18x^{2}-3x-26$

Explanation:

Consider a vertical subtraction process:

$\begin{matrix} 11 x^{2}- 8x-7 \\\underline{ - (-7 x^{2}-5x+19)} \end{matrix}$

Rewrite as the addition of the opposite of the second expression, as follows:

$\begin{matrix} 11 x^{2}- 8x-7 \\\underline{ \; \; 7 x^{2}+5x-19 } \\ 18x^{2}-3x-26 \end{matrix}$

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Example Question #2 : Exponential Operations

Define an operation $\Upsilon$ as follows:

For all real numbers a, b ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $\frac{1}{2} \Upsilon \frac{1}{4}$ .

Possible Answers:

$\frac{1}{32}$

$-\frac{1}{48}$

$\frac{1}{16}$

$\frac{3}{64}$

Correct answer:

$\frac{3}{64}$

Explanation:

$a \Upsilon b = a^{4} - b^{3}$ , so

$\frac{1}{2} \Upsilon \frac{1}{4} = \left ( \frac{1}{2} \right )^{4} - \left (\frac{1}{4} \right )^{3}$

$= \frac{1}{2^{4}} - \frac{1}{4^{3}}$

$= \frac{1}{16} - \frac{1}{64}$

$= \frac{4}{64} - \frac{1}{64}$

$= \frac{3}{64}$

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Example Question #3 : Exponential Operations

Subtract and simplify:

$\left ( 4x^{2}+ 1.2x-7 \right )- \left ( 2x^{2}-3.8x+9\right )$

Possible Answers:

$2x^{2}+ 5x-16$

$2x^{2}-2.6x+2$

The correct answer is not among the other responses.

$2x^{2}+ 5x+2$

$2x^{2}-2.6x-16$

Correct answer:

$2x^{2}+ 5x-16$

Explanation:

Consider a vertical subtraction process:

$\begin{matrix} 4x^{2}+ 1.2x-7\\\underline{ - (2x^{2}-3.8x+9)} \end{matrix}$

Rewrite as the addition of the opposite of the second expression, as follows:

$\begin{matrix} \; \; 4x^{2}+ 1.2x-7\\\underline{ -2x^{2}+3.8x-9}\\ \; 2x^{2}+\; \; 5x-16\end{matrix}$

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Example Question #4 : Exponential Operations

Define an operation $\Upsilon$ as follows:

For all real numbers a, b ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $0.2\; \Upsilon\; 0.1$ .

Possible Answers:

0.0015

0.0006

-0.0015

-0.0084

-0.0006

Correct answer:

0.0006

Explanation:

$a \Upsilon b = a^{4} - b^{3}$

$0.2\; \Upsilon\; 0.1= 0.2^{4} - 0.1^{3}$

= 0.0016 - 0.001

=0.0006

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Example Question #5 : Exponential Operations

Define an operation $\Upsilon$ as follows:

For all real numbers a, b ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $(-4) \Upsilon (-3)$

Possible Answers:

229

283

Correct answer:

283

Explanation:

$a \Upsilon b = a^{4} - b^{3}$

$(-4) \Upsilon (-3) = (-4)^{4}- (-3) ^{3}$

= 256 - (-27)

=283

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Example Question #8 : How To Subtract Exponents

Define as follows:

$f(x) = x^{5} - x^{3}$

Evaluate f(-4) .

Possible Answers:

1,088

960

-1,088

Correct answer:

Explanation:

$f(x) = x^{5} - x^{3}$

$f(-4) = (-4)^{5} - (-4)^{3}= -1,024- (-64) = -960$

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Example Question #4 : Exponents

Simplify the expresseion:

$x^{8} - x^{6}+ x^{2}$

Possible Answers:

$x^{4}$

The expression is already simplified.

Correct answer:

The expression is already simplified.

Explanation:

Variable terms can be combined by adding and subtracting if and only of they are like - that is, if each exponent of each variable is the same. In the given expression, no two exponents are the same. The terms cannot be combined, and the expression is already simplified.

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Example Question #6 : Exponential Operations

Define a function as follows:

$g(x)=\left | x^{5}-x^{2}\right |$

Evaluate g(3) - g(-3) .

Possible Answers:

486

Correct answer:

Explanation:

$g(x)=\left | x^{5}-x^{2}\right |$

$g(3)=\left | 3^{5}-3^{2}\right | = \left | 243-9\right | = \left | 234\right | = 234$

$g(-3)=\left | \left ( -3\right )^{5}- \left ( -3\right )^{2}\right | = \left | -243- 9\right | = \left | -252\right | = 252$

g(3) - g(-3)= 234-252 = -18

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ISEE Upper Level Math : Exponents

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Exponents

Example Question #2 : Exponents

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 #8 : How To Subtract Exponents

Example Question #4 : Exponents

Example Question #6 : Exponential Operations

All ISEE Upper Level Math Resources

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