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ISEE Upper Level Quantitative : How to find the exponent of variables

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

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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #4 : Variables And Exponents

Simplify:

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

Possible Answers:

$\frac{8 y^{12} }{x^{9} }$

$\frac{1}{2y}$

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

$\frac{8x^{9} }{ y^{12}}$

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

Correct answer:

$\frac{8 y^{12} }{x^{9} }$

Explanation:

$\left (\frac{x^{3}}{2y^{4}} \right ) ^{-3} = \left (\frac{2y^{4}}{x^{3}} \right ) ^{3} = \frac{2^{3} \left (y^{4}\right ) ^{3} }{\left (x^{3} \right )^{3}}= \frac{8 y^{4 \cdot 3} }{x^{3\cdot 3} }= \frac{8 y^{12} }{x^{9} }$

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Example Question #4 : Variables And Exponents

y>1

Which is greater?

(a) $y ^{-2}$

(b) $y^{2}$

Possible Answers:

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

Correct answer:

(b) is greater

Explanation:

If y > 1 , then $y ^{2}> 1^{2} = 1$ and $y ^{-2}= \frac{1}{y ^{2}} < \frac{1}{1} = 1$

$y ^{2}> 1> y ^{-2}$ , so by transitivity, $y ^{2}> y ^{-2}$ , and (b) is greater

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Example Question #5 : Variables And Exponents

Expand: $(x + 1)^{10}$

Which is the greater quantity?

(a) The coefficient of $x^{3}$

(b) The coefficient of $x^{7}$

Possible Answers:

(a) is greater.

It is impossible to tell from the information given.

The two quantities are equal.

(b) is greater.

Correct answer:

The two quantities are equal.

Explanation:

By the Binomial Theorem, if $(x + 1)^{n}$ is expanded, the coefficient of $x^{r}$ is

$_{n}\textrm{C} _{r} = \frac{n!}{(n-r)!r!}$ .

(a) Substitute n = 10, r = 3 : The coerfficient of $x^{3}$ is

$_{10}\textrm{C} _{3} = \frac{10!}{(10-3)!3!} = \frac{10!}{7!3!}$ .

(b) Substitute n = 10, r = 7 : The coerfficient of $x^{7}$ is

$_{10}\textrm{C} _{7} = \frac{10!}{(10-7)!7!} = \frac{10!}{3!7!} = \frac{10!}{7!3!}$ .

The two are equal.

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Example Question #2 : How To Find The Exponent Of Variables

Which is greater?

(a) $\left (-2 \right )^{-3}$

(b) $\left ( \frac{1}{2} \right ) ^{3}$

Possible Answers:

It is impossible to tell from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

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

A negative number to an odd power is negative, so the expression in (a) is negative. The expression in (b) is positive since the base is positive. (b) is greater.

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Example Question #4 : How To Find The Exponent Of Variables

x>0

Which is the greater quantity?

(a) $(x + 5)^{3}- (x - 5)^{3}$

(b) 250

Possible Answers:

It is impossble to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(a) is greater.

Explanation:

Simplify the expression in (a):

$(x + 5)^{3}- (x - 5)^{3}$

$=(x^{3}+ 3 \cdot x^{2} \cdot 5 + 3 \cdot x \cdot 5^{2}+ 5^{3}) -(x^{3}- 3 \cdotx^{2} \cdot 5 + 3 \cdot x \cdot 5^{2}- 5^{3})$

$=(x^{3}+15 x^{2}+75 x + 125) - (x^{3}-15 x^{2}+75 x- 125)$

$=x^{3}-x^{3}+15 x^{2}+15 x^{2}+75 x-75 x + 125+125$

$=30 x^{2} + 250$

Since x>0 ,

$(x + 5)^{3}- (x - 5)^{3}=30 x^{2} + 250 > 250$ ,

making (a) greater.

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Example Question #1 : How To Find The Exponent Of Variables

Expand: $(x + 2)^{10}$

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of $x^{2}$

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

The two quantities are equal.

Correct answer:

(b) is greater.

Explanation:

Using the Binomial Theorem, if $(x + 2)^{n}$ is expanded, the $x^{r}$ term is

$_{n}\textrm{C} _{r} \cdot x^{r} \cdot 2 ^{n-r}$

$= \frac{n!}{(n-r)!r!} \cdot 2 ^{n-r} \cdot x^{r}$ .

This makes $\frac{n!}{(n-r)!r!} \cdot 2 ^{n-r}$ the coefficient of $x^{r}$ .

We compare the values of this expression at n = 10 for both r = 1 and r = 2 .

(a) If n = 10 and r = 1 , the coefficient is

$\frac{10!}{(10-1)!1!} \cdot 2 ^{10-1} = \frac{10!}{9!} \cdot 2 ^{9} =10 \cdot 512 =5,120$ .

This is the coefficient of .

(b) If n = 10 and r = 2 , the coefficient is

$\frac{10!}{(10-2)!2!} \cdot 2 ^{10-2} = \frac{10!}{8!2!} \cdot 2 ^{8} = \frac{ 9 \cdot 10}{2} \cdot 256 =11,520$ .

This is the coefficient of $x^{2}$ .

(b) is the greater quantity.

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Example Question #3 : How To Find The Exponent Of Variables

Consider the expression $\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}$

Which is the greater quantity?

(a) The expression evaluated at x = 6, y = 10

(b) The expression evaluated at x = y = 8

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Use the properties of powers to simplify the expression:

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}$

$= \left (x^{2} \right )^{3} y ^{3} \cdot x ^{3}\left ( y ^{2} \right )^{3}$

$= \left (x^{2} \right )^{3} \cdot x ^{3}\cdot y ^{3} \cdot \left ( y ^{2} \right )^{3}$

$= \left (x^{2 \cdot 3} \right ) \cdot x ^{3}\cdot y ^{3} \cdot \left ( y ^{2\cdot 3} \right )$

$= x^{6} \cdot x ^{3}\cdot y ^{3} \cdot y ^{6}$

$= x^{6+3} \cdot y ^{3+ 6}$

$= x^{9} y ^{9} = (xy)^{9}$

(a) If x = 6, y = 10 , then

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}= (xy)^{9} = ( 6 \cdot 10)^{9} = 60^{9}$

(b) If x = y = 8 , then

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}= (xy)^{9} = ( 8 \cdot 8)^{9} = 64^{9}$

(b) is greater.

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Example Question #4 : How To Find The Exponent Of Variables

Which of the following expressions is equivalent to

$\left (100 + \sqrt{x} \right )^{2}$ ?

Possible Answers:

200+x

200-x

None of the other answers is correct.

10,000+x

10,000-x

Correct answer:

None of the other answers is correct.

Explanation:

Use the square of a binomial pattern as follows:

$\left (100 + \sqrt{x} \right )^{2}$

$= 100^{2}+ 2 \cdot 100 \cdot \sqrt {x} +(\sqrt{x})^{2}$

$= 10,000+ 2 00 \sqrt {x} +x$

This expression is not equivalent to any of the choices.

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Example Question #3 : How To Find The Exponent Of Variables

$y = (x+ 3)^{2}$

y = 16z

z = w - 6

Express in terms of .

Possible Answers:

$w = 16 x^{2}+ 96x+ 138$

$w = 16 x^{2}+ 96x+ 105$

$w = 16 x^{2}+ 96x+ 150$

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{25}{16}$

Correct answer:

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

Explanation:

$y = (x+ 3)^{2}$

$= x^{2}+ 2 \cdot 3 \cdot x + 3^{2}$

$= x^{2}+ 6 x + 9$

y = 16z , so

$16z = x^{2}+ 6 x + 9$

$\frac{16z}{16} = \frac{x^{2}+ 6 x + 9}{16}$

$z = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16}$

z = w - 6 , so

w = z+ 6

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16}+6$

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16} + \frac{96}{16}$

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

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Example Question #11 : Variables And Exponents

$a ^{3} = -17$ . Which is the greater quantity?

(a) $a^{6}$

(b)

Possible Answers:

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

By the Power of a Power Principle,

$(a ^{3} )^{2} = a ^{3 \cdot 2 } = a ^{6}$

Therefore,

$a ^{6} = (a ^{3} )^{2} = (-17)^{2} = 17^{2} = 17 \cdot 17 = 289$

It follows that $a ^{6} > - 34$

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the exponent of variables

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #4 : Variables And Exponents

Example Question #4 : Variables And Exponents

Example Question #5 : Variables And Exponents

Example Question #2 : How To Find The Exponent Of Variables

Example Question #4 : How To Find The Exponent Of Variables

Example Question #1 : How To Find The Exponent Of Variables

Example Question #3 : How To Find The Exponent Of Variables

Example Question #4 : How To Find The Exponent Of Variables

Example Question #3 : How To Find The Exponent Of Variables

Example Question #11 : Variables And Exponents

All ISEE Upper Level Quantitative Resources

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