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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 : How To Divide Exponents

Simplify: $\frac{x^{50}}{x^{10}}$

Possible Answers:

$10x^{40}$

$x^{10}$

$x^{5}$

$x^{40}$

$10x^{5}$

Correct answer:

$x^{40}$

Explanation:

Apply the quotient of powers rule.

$\frac{x^{50}}{x^{10}} = x^{50-10} = x^{40}$

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Example Question #2 : How To Divide Exponents

Evaluate:

$\frac{120}{12^{0}}$

Possible Answers:

The expression is undefined

120

Correct answer:

120

Explanation:

Any nonzero number raised to the power of 0 is equal to 1, so $12^{0} = 1$ , and

$\frac{120}{12^{0}} = \frac{120}{1} = 120$

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Example Question #3 : How To Divide Exponents

Evaluate:

$\frac{25}{10 ^{0} -5^{0}}$

Possible Answers:

The expression is undefined.

Correct answer:

The expression is undefined.

Explanation:

Any nonzero number raised to the power of 0 is equal to 1, so $5^{0} = 10 ^{0} =1$ . Therefore,

$\frac{25}{10 ^{0} -5^{0}} = \frac{25}{1-1} = \frac{25}{0}$

However, an expression with a denominator of 0 is undefined, so that is the correct choice.

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Example Question #5 : How To Divide Exponents

Simplify:

$\frac{(2x^4y^{-1})^{-3}}{(3x^3y^2)^{-2}}$

Possible Answers:

$\frac{9y^7}{8x^6}$

$\frac{y^7}{x^6}$

$\frac{y^7}{8x^6}$

$\frac{9y^7}{8x^8}$

$\frac{9y^6}{8x^7}$

Correct answer:

$\frac{9y^7}{8x^6}$

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:

$\frac{(2x^4y^{-1})^{-3}}{(3x^3y^2)^{-2}}=\frac{(3x^3y^2)^{2}}{(2x^4y^{-1})^{3}}$

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.So we have:

$\frac{(3x^3y^2)^{2}}{(2x^4y^{-1})^{3}}=\frac{3^2\times x^{(3\times 2)}\times y^{(2\times 2)}}{2^3\times x^{(4\times 3)}\times y^{(-1\times 3)}}=\frac{9x^6y^4}{8x^{12}y^{-3}}$

Separate the equation:

$\Rightarrow \frac{9x^6y^4}{8x^{12}y^{-3}}=\frac{9}{8}\times \frac{x^6}{x^{12}}\times \frac{y^4}{y^{-3}}$

$=\frac{9}{8}\times x^{6-12}\times y^{4+3}=\frac{9}{8}x^{-6}y^7=\frac{9y^7}{8x^6}$

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Example Question #6 : How To Divide Exponents

Simplify:

$\frac{(x^2y^0)^2}{(x^{-1}y)^{4}}$

Possible Answers:

$\frac{x^8}{y^4}$

$\frac{x^4}{y^4}$

$\frac{x^8}{y}$

$\frac{x^4}{y^8}$

$\frac{x^8}{y^8}$

Correct answer:

$\frac{x^8}{y^4}$

Explanation:

Based on the zero-exponent rule we have:

a^0=1

That means any non-zero number raised to the zero power is equal to . So we can write:

$\frac{(x^2y^0)^2}{(x^{-1}y)^{4}}=\frac{(x^2\times 1)^2}{(x^{-1}y)^{4}}=\frac{(x^2)^2}{(x^{-1}y)^{4}}$

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 negative exponent rule we have:

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

$\frac{(x^2)^2}{(x^{-1}y)^{4}}=\frac{x^{2\times 2}}{(x^{-1\times 4})y^4}=\frac{x^4}{x^{-4}y^4}=\frac{x^{4+4}}{y^4}=\frac{x^8}{y^4}$

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Example Question #7 : How To Divide Exponents

Evaluate:

$\frac{(11-7)^2}{(9-7)^3}$

Possible Answers:

Correct answer:

Explanation:

First we need to subtract the numbers in parentheses. So we can write:

$\frac{(11-7)^2}{(9-7)^3}=\frac{4^2}{2^3}=\frac{4\times 4}{2\times 2\times 2}=\frac{16}8{}=2$

We can alternatively write:

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

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Example Question #8 : How To Divide Exponents

Simplify:

$\frac{6x^6}{2x^3}$

Possible Answers:

3x^3

3x^2

3x^4

2x^3

Correct answer:

3x^3

Explanation:

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

We should separate the fraction and apply the quotient rule for exponents:

$\frac{6x^6}{2x^3}=\frac{6}{2}\times \frac{x^6}{x^3}=3\times x^{6-3}$

$\Rightarrow \frac{6x^6}{2x^3}=3x^3$

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Example Question #9 : How To Divide Exponents

Simplify:

$\frac{4x^8}{2x^0}$

Possible Answers:

x^8

x^7

2x^9

2x^7

2x^8

Correct answer:

2x^8

Explanation:

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

We also know that any non-zero number raised to the power of is .

$\frac{4x^8}{2x^0}=\frac{4}{2}\times \frac{x^8}{1}=2x^8$

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Example Question #10 : How To Divide Exponents

Simplify:

$\frac{6x^4-6}{3x^2+3x^0}$

Possible Answers:

2(x-1)

x^2-1

2(x^2-1)

2(x+1)

2(x^2+1)

Correct answer:

2(x^2-1)

Explanation:

We know that any non-zero number raised to the power of is equal to .

$\frac{6x^4-6}{3x^2+3x^0}=\frac{6(x^4-1)}{3(x^2+x^0)}=\frac{6(x^4-1)}{3(x^2+1)}$

Now factor:

$\frac{6(x^4-1)}{3(x^2+1)}=\frac{6}{3}\frac{(x^2-1)(x^2+1)}{x^2+1}=2(x^2-1)$

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Example Question #31 : Exponential Operations

Evaluate:

$\frac{4^0}{2\times 5^0-12^0}$

Possible Answers:

The expression is undefined.

$-\frac{2}{5}$

Correct answer:

Explanation:

We know that any non-zero number raised to the power of is equal to .

$\frac{4^0}{2\times 5^0-12^0}=\frac{1}{2\times 1-1}=\frac{1}{2-1}=1$

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All ISEE Upper Level Math Resources

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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 : How To Divide Exponents

Example Question #2 : How To Divide Exponents

Example Question #3 : How To Divide Exponents

Example Question #5 : How To Divide Exponents

Example Question #6 : How To Divide Exponents

Example Question #7 : How To Divide Exponents

Example Question #8 : How To Divide Exponents

Example Question #9 : How To Divide Exponents

Example Question #10 : How To Divide Exponents

Example Question #31 : Exponential Operations

All ISEE Upper Level Math Resources

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