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ISEE Upper Level Math : Exponents

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All ISEE Upper Level Math Resources

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Example Questions

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Example Question #61 : Numbers And Operations

Define a function $g(x)=\left | x^{5}-x^{3}\right |$ .

Evaluate g(4)-g(-4)

Possible Answers:

-1,920

960

1,920

Correct answer:

Explanation:

$g(x)=\left | x^{5}-x^{3}\right |$

$g(4)=\left | 4^{5}-4^{3}\right | = \left | 1,024-64\right | = \left | 960\right |= 960$

$g(4)=\left | \left (-4 \right )^{5}- \left (-4 \right )^{3}\right | = \left | -1,024-\left ( - 64 \right )\right | = \left | -960\right |= 960$

g(4)-g(-4) = 960 - 960 = 0

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Example Question #11 : How To Subtract Exponents

Define $f(x)= x^{6} + x^{3}$ and $g(x) = x^{6}-x^{3}$ .

Evaluate $\left (\frac{f}{g} \right ) (-1)$ .

Possible Answers:

$\frac{1}{2}$

Undefined

Correct answer:

Explanation:

$\left (\frac{f}{g} \right ) (x) = \frac{f(x)}{g(x)}$ , by definition, so

$\left (\frac{f}{g} \right ) (-1)= \frac{f(-1)}{g(-1)}$

Evaluate f(-1) and g(-1) separately:

$f(x)= x^{6} + x^{3}$

$f(-1)= (-1)^{6} + (-1)^{3} = 1+(-1)= 0$

$g(x) = x^{6}-x^{3}$

$g(-1)= (-1)^{6} - (-1)^{3} = 1-(-1)= 2$

$\left (\frac{f}{g} \right ) (-1)= \frac{f(-1)}{g(-1)} = \frac{0}{2}= 0$

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Example Question #61 : Numbers And Operations

Define as follows:

$f(x) = x^{6} - x^{5}$

Evaluate f (-3) .

Possible Answers:

486

972

Correct answer:

972

Explanation:

$f(x) = x^{6} - x^{5}$

$f(x) = (-3)^{6} - (-3)^{5} = 729-(-243)= 972$

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Example Question #1 : How To Multiply Exponents

$\dpi{100} (7-4)^{3}=$

Possible Answers:

$\dpi{100} 4$

$\dpi{100} 27$

$\dpi{100} 3$

$\dpi{100} 9$

Correct answer:

$\dpi{100} 27$

Explanation:

Subtract the numbers inside the parentheses first. This leaves you with 3, which you then raise to the 3rd power:

$\dpi{100} 3\times 3\times 3=27$

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Example Question #2 : How To Multiply Exponents

Evaluate:

$\frac{1^{0}+2 ^{0}+3 ^{0}}{6^{0}}$

Possible Answers:

$\frac{1}{6}$

The expression is undefined.

Correct answer:

Explanation:

Any nonzero number taken to the power of 0 is equal to 1, so

$\frac{1^{0}+2 ^{0}+3 ^{0}}{6^{0}} = \frac{1+1+1}{1} = \frac{3}{1} = 3$

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Example Question #1 : How To Multiply Exponents

Multiply:

$(4r^{2} - 7) (5r^{2}+ 8 )$

Possible Answers:

$20r^{4} -67r^{2} -56$

$20r^{4} +3r^{2} -56$

$20r^{4} -56$

$20r^{4} -3r^{2} -56$

$20r^{4} +67r^{2} -56$

Correct answer:

$20r^{4} -3r^{2} -56$

Explanation:

Use the FOIL method:

$(4r^{2} - 7) (5r^{2}+ 8 )$

$=4r^{2} \cdot 5r^{2} + 4r^{2} \cdot 8 -7 \cdot 5r^{2} -7 \cdot 8$

$=4\cdot 5 \cdot r^{2}\cdot r^{2} + 4\cdot 8\cdot r^{2} -7 \cdot 5\cdot r^{2} -7 \cdot 8$

$=20 \cdot r^{2+2} + 32r^{2} -35r^{2} -56$

$=20r^{4} +\left ( 32-35 \right )r^{2} -56$

$=20r^{4} -3r^{2} -56$

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Example Question #4 : How To Multiply Exponents

Evaluate:

$-x(x^3)^{2}$

Possible Answers:

-x^7

-x^8

x^7

-x^5

-x^6

Correct answer:

-x^7

Explanation:

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. In addition, based on the product rule for exponents in order to multiply two exponential terms with the same base, we need to add their exponents:

$(x)^a(x)^b=x^{a+b}$

So we can write:

$-x(x^3)^{2}=-x(x^{3\times 2})=-x(x^6)=-x^{1+6}=-x^7$

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Example Question #5 : How To Multiply Exponents

Simplify:

$(6a^3)(\frac{1}{3}a^2)$

Possible Answers:

a^6

2a^6

2a^4

a^5

2a^5

Correct answer:

2a^5

Explanation:

Based on the product rule for exponents in order to multiply two exponential terms with the same base, add their exponents:

$(x)^a(x)^b=x^{a+b}$

So we can write:

$(6a^3)\left(\frac{1}{3}a^2\right)=\left(\frac{6}{3}\right)\times a^{3+2}=2a^5$

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Example Question #6 : How To Multiply Exponents

Evaluate:

$\frac{4x^2(x^3)^{3}}{8x^7}$

Possible Answers:

$\frac{1}{2}x^3$

x^4

x^3

$\frac{1}{2}x^4$

$\frac{1}{2}x^5$

Correct answer:

$\frac{1}{2}x^4$

Explanation:

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. In addition, based on the product rule for exponents, in order to multiply two exponential terms with the same base we need to add their exponents:

$(x)^a(x)^b=x^{a+b}$

So we can write:

$\frac{4x^2(x^3)^{3}}{8x^7}=\frac{4x^2(x^{3\times 3})}{8x^7}=\frac{4x^2\times x^9}{8x^7}=\left(\frac{4}{8}\right)\frac{x^{2+9}}{x^7}=\frac{1}{2}\cdot \frac{x^{11}}{x^7}$

in order to divide two exponents with the same base, we can keep the base and subtract the powers. So we get:

$\frac{4x^2(x^3)^{3}}{8x^7}=\frac{1}{2}\cdot \frac{x^{11}}{x^7}=\frac{1}{2}x^{11-7}=\frac{1}{2}x^4$

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Example Question #7 : How To Multiply Exponents

Evaluate:

$-2x^2(x^{-3})^{3}$

Possible Answers:

$2x^{7}$

$2x^{-7}$

$-2x^{7}$

$-x^{-7}$

$-2x^{-7}$

Correct answer:

$-2x^{-7}$

Explanation:

Based on the negative exponent rule we have:

$a^{-n}=\frac{1}{a^n}$

which says negative exponents in the numerator get moved to the denominator and become positive exponents. And negative exponents in the denominator get moved to the numerator and become positive exponents. So we can write:

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

In addition, based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. We also know that when a fraction is raised to a power, the numerator and the denominator are both raised to that power. So we can write:

$-2x^2(x^{-3})^{3}=-2x^2\left(\frac{1}{x^3}\right)^{3}=-2x^2\times \frac{1^3}{x^{3\times 3}}$

$=-2x^2\times \frac{1}{x^9}$

$=\frac{-2x^2}{x^9}$

$=-2x^{2-9}$

$=-2x^{-7}$

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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 #61 : Numbers And Operations

Example Question #11 : How To Subtract Exponents

Example Question #61 : Numbers And Operations

Example Question #1 : How To Multiply Exponents

Example Question #2 : How To Multiply Exponents

Example Question #1 : How To Multiply Exponents

Example Question #4 : How To Multiply Exponents

Example Question #5 : How To Multiply Exponents

Example Question #6 : How To Multiply Exponents

Example Question #7 : How To Multiply Exponents

All ISEE Upper Level Math Resources

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