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ISEE Upper Level Math : Variables

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 : How To Divide Exponential Variables

Simplify:

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

Possible Answers:

18x^5

$\frac{9}{x^5}$

$\frac{18}{x^4}$

9x^5

$\frac{18}{x^5}$

Correct answer:

$\frac{18}{x^5}$

Explanation:

In the denominator, apply the exponent to the terms in parentheses.

$\frac{2x^{-7}}{(3x)^{-2}}=\frac{2x^{-7}}{(3^{-2}) (x^{-2})}$

Terms with negative exponents can be inverted within the fraction to eliminate the negative.

$\frac{2x^{-7}}{(3^{-2}) (x^{-2})}=\frac{2(3^2)(x^2)}{x^7}$

Simplify by canceling variables and mulitplying.

$\frac{2(3^2)(x^2)}{x^7}=\frac{18x^2}{x^7}=\frac{18}{x^5}$

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Example Question #7 : How To Divide Exponential Variables

Simplify:

$\frac{8x^{-2a}y^{2a}}{4x^ay^{-2a}}$

Possible Answers:

$y^{4a}}{x^{3a}$

$\frac{y^{4a}}{x^{3a}}$

$\frac{2y^{4a}}{x^{3a}}$

$2y^{4a}}{x^{3a}$

$\frac{2y^{3a}}{x^{3a}}$

Correct answer:

$\frac{2y^{4a}}{x^{3a}}$

Explanation:

Break up the fraction into components with like-terms.

$\frac{8x^{-2a}y^{2a}}{4x^ay^{-2a}}=\frac{8}{4}\times \frac{x^{-2a}}{x^a}\times \frac{y^{2a}}{y^{-2a}}$

Simplify by combining exponents. Remember that exponents in the denominator will become negative.

$2\times x^{(-2a-a)}\times y^{(2a+2a)}$

$2x^{-3a}y^{4a}$

Finally, put any negative exponent terms into the denominator.

$\frac{2y^{4a}}{x^{3a}}$

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Example Question #2 : How To Divide Exponential Variables

Simplify:

$\frac{(2xy)^{6}}{2x^{2} y^{8}}$

Possible Answers:

$32 x^{4} y^{2}$

$\frac{32y^{2} }{ x^{4}}$

$\frac{32 }{ x^{4}y^{2}}$

$\frac{32 x^{4} }{ y^{2}}$

Correct answer:

$\frac{32 x^{4} }{ y^{2}}$

Explanation:

$\frac{(2xy)^{6}}{2x^{2} y^{8}}$

$= \frac{2^{6} \cdot x^{6}y^{6}}{2x^{2} y^{8}}$

$= \frac{64 x^{6-2}}{2 y^{8-6}}$

$= \frac{32 x^{4} }{ y^{2}}$

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Example Question #9 : How To Divide Exponential Variables

Divide the following:

$x^8 \div x^2$

Possible Answers:

x^4

x^3

x^6

x^2

$x^{10}$

Correct answer:

x^6

Explanation:

To divide variables with exponents, we will use the following formula:

$x^a \div x^b = x^{a-b}$

Now, let’s combine the following:

$x^8 \cdot x^2 = x^{8-2}$

$x^8 \cdot x^2 = x^{6}$

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Example Question #10 : How To Divide Exponential Variables

Divide the following:

$\frac{a^2 b^4}{a b^2}$

Possible Answers:

a^3b^6

a^3b

a^2b

ab^2

Correct answer:

ab^2

Explanation:

When dividing variables, we can only divide like variable.

Also, when dividing variables with exponents, we subtract the exponents. In other words:

$\frac{a^x}{a^y} = a^{x-y}$

So, we get

$\frac{a^2 b^4}{a b^2}$

$\frac{a^2b^4}{a^1b^2}$

$a^{2-1}b^{4-2}$

a^1b^2

ab^2

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Example Question #131 : Variables

Simplify:

$y^{12} \div y^2$

Possible Answers:

y^4

$y^{10}$

$y^{14}$

y^6

y^8

Correct answer:

$y^{10}$

Explanation:

To simplify, we will use the following formula:

$x^a \div x^b = x^{a-b}$

So, given the problem, we get

$y^{12} \div y^2 = y^{12-2}$

$y^{12} \div y^2 = y^{10}$

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Example Question #132 : Variables

Divide the following:

$\frac{h^{10}}{h^2}$

Possible Answers:

h^1

h^8

$h^{12}$

h^5

$h^{20}$

Correct answer:

h^8

Explanation:

To divide like terms with exponents, we will use the following formula:

$\frac{a^x}{a^y} = a^{x-y}$

So, we get

$\frac{h^{10}}{h^2}$

$h^{10-2}$

h^8

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Example Question #133 : Variables

Divide the following:

$\frac{y^9}{y^3}$

Possible Answers:

y^2

y^3

y^8

y^6

2y^2

Correct answer:

y^6

Explanation:

To divide like variables with exponents, we will use the following formula:

$\frac{a^x}{a^y} = a^{x-y}$

So, we get

$\frac{y^9}{y^3} = y^{9-3}$

$\frac{y^9}{y^3} = y^6$

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Example Question #134 : Variables

Divide the following::

$p^9 \div p^3$

Possible Answers:

p^1

p^3

$p^{12}$

p^6

$p^{93}$

Correct answer:

p^6

Explanation:

To divide variables with exponents, we will use the following formula:

$x^a \div x^b = x^{a-b}$

Now, let’s divide the following:

$p^9 \div p^3 = p^{9-3}$

$p^9 \div p^3 = p^{6}$

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Example Question #135 : Variables

Solve the following:

$\frac{18x^4}{9x}$

Possible Answers:

9x^3

9x^2

2x^4

2x^3

Correct answer:

2x^3

Explanation:

To divide like variables with exponents, we will use the following formula:

$\frac{a^x}{a^y} = a^{x-y}$

Also, we divide the coefficients like normal.

So, we get

$\frac{18x^4}{9x}$

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

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

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

$2x^{4-1}$

2x^3

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Variables

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : How To Divide Exponential Variables

Example Question #7 : How To Divide Exponential Variables

Example Question #2 : How To Divide Exponential Variables

Example Question #9 : How To Divide Exponential Variables

Example Question #10 : How To Divide Exponential Variables

Example Question #131 : Variables

Example Question #132 : Variables

Example Question #133 : Variables

Example Question #134 : Variables

Example Question #135 : Variables

All ISEE Upper Level Math Resources

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