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ISEE Upper Level Quantitative : Variables

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

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : How To Divide Exponential Variables

Simplify:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$ $\displaystyle \frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

Possible Answers:

$\frac{2}{x ^{2}y ^{2} }$ $\displaystyle \frac{2}{x ^{2}y ^{2} }$

$\frac{1 }{2 x ^{2}y ^{2}}$ $\displaystyle \frac{1 }{2 x ^{2}y ^{2}}$

$\frac{x ^{2} }{2y ^{2} }$ $\displaystyle \frac{x ^{2} }{2y ^{2} }$

$\frac{x ^{2}y ^{2} }{2 }$ $\displaystyle \frac{x ^{2}y ^{2} }{2 }$

$\frac{y ^{2} }{2x ^{2} }$ $\displaystyle \frac{y ^{2} }{2x ^{2} }$

Correct answer:

$\frac{x ^{2} }{2y ^{2} }$ $\displaystyle \frac{x ^{2} }{2y ^{2} }$

Explanation:

Break the fraction up and apply the quotient of powers rule:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$ $\displaystyle \frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

$= \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$ $\displaystyle = \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$

$= \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$ $\displaystyle = \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$

$= \frac{1 }{2} \cdot x ^{2} \cdot y ^{-2}$ $\displaystyle = \frac{1 }{2} \cdot x ^{2} \cdot y ^{-2}$

$= \frac{1 }{2} \cdot x ^{2} \cdot \frac{1}{y ^{2}}$ $\displaystyle = \frac{1 }{2} \cdot x ^{2} \cdot \frac{1}{y ^{2}}$

$= \frac{x ^{2} }{2y ^{2} }$ $\displaystyle = \frac{x ^{2} }{2y ^{2} }$

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Example Question #261 : Algebraic Concepts

Simplify: $\frac{16x^4}{12x^2}$ $\displaystyle \frac{16x^4}{12x^2}$

Possible Answers:

$\frac{4}{3}$ $\displaystyle \frac{4}{3}$

16x $\displaystyle 16x$

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

$\frac{4}{3x^2}$ $\displaystyle \frac{4}{3x^2}$

$\frac{4x^2}{3}$ $\displaystyle \frac{4x^2}{3}$

Correct answer:

$\frac{4x^2}{3}$ $\displaystyle \frac{4x^2}{3}$

Explanation:

To simplify this expression, look at the like terms separately. First, simplify $\frac{16}{12}$ $\displaystyle \frac{16}{12}$ . This becomes $\frac{4}{3}$ $\displaystyle \frac{4}{3}$ . Then, deal with the $\frac{x^4}{x^2}$ $\displaystyle \frac{x^4}{x^2}$ . Since the bases are the same and you're dividing, you can subtract exponents. This gives you x^2. $\displaystyle x^2.$ Since the exponent is positive, you put in the numerator. This gives you a final answer of $\frac{4x^2}{3}$ $\displaystyle \frac{4x^2}{3}$ .

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Example Question #5 : How To Divide Exponential Variables

$\displaystyle x$ is a negative number.

Which is the greater quantity?

(a) The reciprocal of $x^{7}$ $\displaystyle x^{7}$

(b) The reciprocal of $x^{8}$ $\displaystyle x^{8}$

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

It is impossible to determine which is greater from the information given

Correct answer:

(b) is the greater quantity

Explanation:

A negative number raised to an odd power is negative; a negative number raised to an even power is positive. It follows that $x^{7}$ $\displaystyle x^{7}$ is negative and $x^{8}$ $\displaystyle x^{8}$ is positive. Also, the reciprocal of a nonzero number assumes the same sign as the number itself, so the reciprocal of $x^{8}$ $\displaystyle x^{8}$ is positive and that of $x^{7}$ $\displaystyle x^{7}$ is negative. It follows that the reciprocal of $x^{8}$ $\displaystyle x^{8}$ is the greater of the two.

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Example Question #2 : How To Divide Exponential Variables

Simplify:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$ $\displaystyle \frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

Possible Answers:

$\frac{y ^{2} }{2x ^{2} }$ $\displaystyle \frac{y ^{2} }{2x ^{2} }$

$\frac{2}{x ^{2}y ^{2} }$ $\displaystyle \frac{2}{x ^{2}y ^{2} }$

$\frac{x ^{2} }{2y ^{2} }$ $\displaystyle \frac{x ^{2} }{2y ^{2} }$

$\frac{x ^{2}y ^{2} }{2 }$ $\displaystyle \frac{x ^{2}y ^{2} }{2 }$

$\frac{1 }{2 x ^{2}y ^{2}}$ $\displaystyle \frac{1 }{2 x ^{2}y ^{2}}$

Correct answer:

$\frac{x ^{2} }{2y ^{2} }$ $\displaystyle \frac{x ^{2} }{2y ^{2} }$

Explanation:

Break the fraction up and apply the quotient of powers rule:

$\frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$ $\displaystyle \frac{10 x ^{-6} y ^{-4}}{20x ^{-8}y^{-2}}$

$= \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$ $\displaystyle = \frac{10 }{20} \cdot \frac{ x ^{-6} }{x ^{-8}} \cdot \frac{y ^{-4}}{y^{-2}}$

$= \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$ $\displaystyle = \frac{1 }{2} \cdot x ^{-6- (-8)} \cdot y ^{-4- (-2)}$

$= \frac{1 }{2} \cdot x ^{2} \cdot y ^{-2}$ $\displaystyle = \frac{1 }{2} \cdot x ^{2} \cdot y ^{-2}$

$= \frac{1 }{2} \cdot x ^{2} \cdot \frac{1}{y ^{2}}$ $\displaystyle = \frac{1 }{2} \cdot x ^{2} \cdot \frac{1}{y ^{2}}$

$= \frac{x ^{2} }{2y ^{2} }$ $\displaystyle = \frac{x ^{2} }{2y ^{2} }$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Variables

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #2 : How To Divide Exponential Variables

Example Question #261 : Algebraic Concepts

Example Question #5 : How To Divide Exponential Variables

Example Question #2 : How To Divide Exponential Variables

All ISEE Upper Level Quantitative Resources

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