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ISEE Upper Level Math : Exponents

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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 : Exponential Operations

Simplify:

$\displaystyle 3a^3-a^2+4a^2-8a^3$

Possible Answers:

$\displaystyle a(-5a+3)$

$\displaystyle a^2(5a+3)$

$\displaystyle a^2(-5a+2)$

$\displaystyle a^2(5a-3)$

$\displaystyle a^2(-5a+3)$

Correct answer:

$\displaystyle a^2(-5a+3)$

Explanation:

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

$\displaystyle 3a^3-a^2+4a^2-8a^3=(3a^3-8a^3)+(4a^2-a^2)$

$\displaystyle =-5a^3+3a^2=a^2(-5a+3)$

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Example Question #2 : Exponents

Evaluate:

$\displaystyle 5^2+5^3-7^2-3(7^2)-4(5^3)$

Possible Answers:

$\displaystyle -546$

550 $\displaystyle 550$

$\displaystyle -500$

546 $\displaystyle 546$

$\displaystyle -550$

Correct answer:

$\displaystyle -546$

Explanation:

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

$\displaystyle 5^2+5^3-7^2-3(7^2)-4(5^3)=5^2+(5^3-4(5^3))-(7^2+3(7^2))$

$\displaystyle =5^2+-3(5^3)-4(7^2)$

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

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

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

$\displaystyle =25-375-196=-546$

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Example Question #3 : Exponents

Subtract and simplify:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

Consider a vertical subtraction process:

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

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

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

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Example Question #4 : Exponents

Define an operation $\Upsilon$ $\displaystyle \Upsilon$ as follows:

For all real numbers a, b $\displaystyle a, b$,

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

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

Possible Answers:

$\displaystyle 0$

$\frac{1}{32}$ $\displaystyle \frac{1}{32}$

$-\frac{1}{48}$ $\displaystyle -\frac{1}{48}$

$\frac{3}{64}$ $\displaystyle \frac{3}{64}$

$\frac{1}{16}$ $\displaystyle \frac{1}{16}$

Correct answer:

$\frac{3}{64}$ $\displaystyle \frac{3}{64}$

Explanation:

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

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

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

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

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

$= \frac{3}{64}$ $\displaystyle = \frac{3}{64}$

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Example Question #3 : Exponents

Subtract and simplify:

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

Possible Answers:

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

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

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

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

The correct answer is not among the other responses.

Correct answer:

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

Explanation:

Consider a vertical subtraction process:

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

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

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

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Example Question #59 : Numbers And Operations

Define an operation $\Upsilon$ $\displaystyle \Upsilon$ as follows:

For all real numbers a, b $\displaystyle a, b$,

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

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

Possible Answers:

-0.0006 $\displaystyle -0.0006$

0.0015 $\displaystyle 0.0015$

0.0006 $\displaystyle 0.0006$

-0.0015 $\displaystyle -0.0015$

-0.0084 $\displaystyle -0.0084$

Correct answer:

0.0006 $\displaystyle 0.0006$

Explanation:

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

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

$\displaystyle = 0.0016 - 0.001$

=0.0006 $\displaystyle =0.0006$

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Example Question #2 : Exponents

Define an operation $\Upsilon$ $\displaystyle \Upsilon$ as follows:

For all real numbers a, b $\displaystyle a, b$,

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

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

Possible Answers:

$\displaystyle -229$

283 $\displaystyle 283$

$\displaystyle -283$

229 $\displaystyle 229$

$\displaystyle -7$

Correct answer:

283 $\displaystyle 283$

Explanation:

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

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

$\displaystyle = 256 - (-27)$

=283 $\displaystyle =283$

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Example Question #6 : Exponents

Define $\displaystyle f$ as follows:

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

Evaluate f(-4) $\displaystyle f(-4)$.

Possible Answers:

-1,088 $\displaystyle -1,088$

$\displaystyle -960$

1,088 $\displaystyle 1,088$

960 $\displaystyle 960$

$\displaystyle -16$

Correct answer:

$\displaystyle -960$

Explanation:

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

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

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Example Question #7 : Exponents

Simplify the expresseion:

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

Possible Answers:

$x^{4}$ $\displaystyle x^{4}$

$\displaystyle 1$

$\displaystyle 0$

$\displaystyle x$

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 #3 : Exponents

Define a function $\displaystyle g$ as follows:

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

Evaluate $\displaystyle g(3) - g(-3)$.

Possible Answers:

$\displaystyle -18$

$\displaystyle 18$

$\displaystyle 0$

486 $\displaystyle 486$

$\displaystyle -486$

Correct answer:

$\displaystyle -18$

Explanation:

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

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

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

$\displaystyle g(3) - g(-3)= 234-252 = -18$

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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 : Exponential Operations

Example Question #2 : Exponents

Example Question #3 : Exponents

Example Question #4 : Exponents

Example Question #3 : Exponents

Example Question #59 : Numbers And Operations

Example Question #2 : Exponents

Example Question #6 : Exponents

Example Question #7 : Exponents

Example Question #3 : Exponents

All ISEE Upper Level Math Resources

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