ISEE Upper Level Quantitative : How to divide exponential variables

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

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Example Questions

Example Question #261 : Algebraic Concepts

Half of one hundred divided by five and multiplied by one-tenth is __________.

Possible Answers:

10

5

2

1

Correct answer:

1

Explanation:

Let's take this step by step. "Half of one hundred" is 100/2 = 50. Then "half of one hundred divided by five" is 50/5 = 10. "Multiplied by one-tenth" really is the same as dividing by ten, so the last step gives us 10/10 = 1.

Example Question #1 : How To Divide Exponential Variables

Simplify:  \(\displaystyle \frac{8x^{-6}}{2x^{-3}}\)

Possible Answers:

\(\displaystyle \frac{x^{3}}{64}\)

\(\displaystyle 4x^{3}\)

\(\displaystyle \frac{4}{x^{3}}\)

\(\displaystyle 64x^{3}\)

\(\displaystyle \frac{x^{3}}{4}\)

Correct answer:

\(\displaystyle \frac{4}{x^{3}}\)

Explanation:

\(\displaystyle \frac{8x^{-6}}{2x^{-3}} = \frac{8}{2} \cdot \frac{x^{-6}}{x^{-3}} = 4 \cdot x ^{-6 - (-3)} = 4 \cdot x ^{-3} =4 \cdot \frac{1}{x^{3}} = \frac{4}{x^{3}}\)

Example Question #263 : Algebraic Concepts

Simplify: 

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

Possible Answers:

\(\displaystyle \frac{1 }{2 x ^{2}y ^{2}}\)

\(\displaystyle \frac{2}{x ^{2}y ^{2} }\)

\(\displaystyle \frac{x ^{2}y ^{2} }{2 }\)

\(\displaystyle \frac{x ^{2} }{2y ^{2} }\)

\(\displaystyle \frac{y ^{2} }{2x ^{2} }\)

Correct answer:

\(\displaystyle \frac{x ^{2} }{2y ^{2} }\)

Explanation:

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

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

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

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

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

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

\(\displaystyle = \frac{x ^{2} }{2y ^{2} }\)

Example Question #264 : Algebraic Concepts

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

Possible Answers:

\(\displaystyle \frac{4}{3x^2}\)

\(\displaystyle \frac{4x^2}{3}\)

\(\displaystyle 16x\)

\(\displaystyle \frac{4}{3}\)

\(\displaystyle \frac{x^2}{3}\)

Correct answer:

\(\displaystyle \frac{4x^2}{3}\)

Explanation:

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

Example Question #5 : How To Divide Exponential Variables

\(\displaystyle x\) is a negative number.

Which is the greater quantity?

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

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

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 \(\displaystyle x^{7}\) is negative and \(\displaystyle x^{8}\) is positive. Also, the reciprocal of a nonzero number assumes the same sign as the number itself, so the reciprocal of \(\displaystyle x^{8}\) is positive and that of \(\displaystyle x^{7}\) is negative. It follows that the reciprocal of \(\displaystyle x^{8}\) is the greater of the two.

Example Question #2 : How To Divide Exponential Variables

Simplify: 

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

Possible Answers:

\(\displaystyle \frac{y ^{2} }{2x ^{2} }\)

\(\displaystyle \frac{2}{x ^{2}y ^{2} }\)

\(\displaystyle \frac{x ^{2} }{2y ^{2} }\)

\(\displaystyle \frac{x ^{2}y ^{2} }{2 }\)

\(\displaystyle \frac{1 }{2 x ^{2}y ^{2}}\)

Correct answer:

\(\displaystyle \frac{x ^{2} }{2y ^{2} }\)

Explanation:

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

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

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

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

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

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

\(\displaystyle = \frac{x ^{2} }{2y ^{2} }\)

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