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ISEE Upper Level Math : Numbers and Operations

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

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Example Question #1 : Exponential Operations

Define as follows:

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

Evaluate f(-4) .

Possible Answers:

-1,088

1,088

960

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 #1 : How To Subtract Exponents

Simplify the expresseion:

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

Possible Answers:

The expression is already simplified.

$x^{4}$

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 #1 : How To Subtract Exponents

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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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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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Numbers and Operations

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 #1 : How To Subtract Exponents

Example Question #1 : How To Subtract Exponents

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

All ISEE Upper Level Math Resources

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