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ISEE Upper Level Math : Variables and Exponents

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Example Questions

Example Question #11 : Variables And Exponents

Simplify:

$\left (6x ^{-5 } \right )^{-3}$

Possible Answers:

$-18x^{15}$

$-\frac{x^{2}}{216}$

$\frac{x^{15}}{216}$

$216x^{2}$

$-18x^{2}$

Correct answer:

$\frac{x^{15}}{216}$

Explanation:

Apply the power of a product property:

$\left (6x ^{-5 } \right )^{-3}$

$= 6 ^{-3}\cdot \left (x ^{-5 } \right )^{-3}$

$=\frac{1}{216}\cdot x ^{-5 \cdot (-3) }$

$=\frac{1}{216}\cdot x ^{15 }$

$=\frac{x ^{15 }}{216}$

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Example Question #12 : Variables And Exponents

What is the coefficient of $x^{5}$ in the expansion of $(x+1)^{12}$ .

Possible Answers:

7,920

9,504

792

Correct answer:

792

Explanation:

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

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

Substitute n = 12, r = 5 : The coefficient of $x^{5}$ is:

$_{12}\textrm{C} _{5} = \frac{12!}{(12-5)!5!} = \frac{12!}{7!5!}$

$= \frac{12 \cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}$

$= \frac{12 \cdot 11\cdot 10\cdot 9\cdot 8}{ 5\cdot 4\cdot 3\cdot 2\cdot 1} = \frac{11\cdot 9\cdot 8}{ 1} = 792$

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Example Question #11 : Variables And Exponents

Simplify the expression: $\left [\left ( x ^{3} \right )^{3} \right ]^{3 }$

Possible Answers:

$x^ {18}$

$x^{27}$

$x ^{9}$

$27x^{3}$

729x

Correct answer:

$x^{27}$

Explanation:

Apply the power of a power property twice:

$\left [\left ( x ^{3} \right )^{3} \right ]^{3 } = ( x ^{3\; \cdot \; 3 } )^{3 } = ( x ^{9 } )^{3 } = x ^{9\cdot 3} =x ^{27}$

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Example Question #12 : Variables And Exponents

What is the coefficient of $x^{4}$ in the expansion of $(2x + 3)^{6}$ ?

Possible Answers:

240

2,160

135

5,320

Correct answer:

2,160

Explanation:

By the Binomial Theorem, the $x^{r}$ term of $(ax + b)^{n}$ is

$_{n}\textrm{C} _{r} b^{n-r}\left ( ax \right )^{r}= \frac{n!}{(n-r)!r!} a^{r} b^{n-r} x^{r}$ ,

making the coefficient of $x^{r}$

$\frac{n!}{(n-r)!r!} a^{r} b^{n-r}$ .

We can set a = 2,b=3,n=6,r=4 in this expression:

$\frac{6!}{(6-4)!4!} \cdot 2^{4} \cdot 3^{6-4}$

$=\frac{6!}{2!4!}\cdot 2^{4}\cdot 3^{2}$

$=\frac{720}{2\cdot 24}\cdot 16\cdot 9$

=2,160

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

What is the coefficient of $x^{8}$ in the expansion of $(x + 3)^{10}$ ?

Possible Answers:

405

3,240

135

Correct answer:

405

Explanation:

By the Binomial Theorem, the $x^{r}$ term of $(x + b)^{n}$ is

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

Substitute b = 3, n = 10, r= 8 and this becomes

$\frac{10!}{(10-8)!8!} 3^{10-8} x^{8}$ .

The coefficient is

$\frac{10!}{(10-8)!8!} \cdot 3^{10-8} = \frac{10!}{2!8!}\cdot 3^{2} = \frac{9 \cdot 10}{2}\cdot 9 = 405$ .

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Example Question #13 : Variables And Exponents

Evaluate:

$\left [ (x^4)^4 \right ]^5$

Possible Answers:

$x^{40}$

$x^{80}$

$x^{60}$

$x^{12}$

$x^{16}$

Correct answer:

$x^{80}$

Explanation:

We need to apply the power of power rule twice:

$\left [ (x^4)^4 \right ]^5=(x^{4\times 4})^5=(x^{16})^5=x^{16\times 5}=x^{80}$

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Example Question #14 : Variables And Exponents

$(x^{-4})^{a}=x^{8}$

Solve for .

Possible Answers:

Correct answer:

Explanation:

Based on the power of a product rule we have:

$(x^{-4})^{a}=x^{8}\Rightarrow x^{-4\times a}=x^8$

The bases are the same, so we can write:

$-4a=8\Rightarrow a=\frac{8}{-4}=-2$

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Example Question #5 : How To Find The Exponent Of Variables

Simplify:

$(\frac{2x^5}{3y^2})^{-4}$

Possible Answers:

$\frac{y^8}{16x^{20}}$

$\frac{81y^8}{x^{20}}$

$\frac{81y^8}{16x^{16}}$

$\frac{81y^9}{16x^{20}}$

$\frac{81y^8}{16x^{20}}$

Correct answer:

$\frac{81y^8}{16x^{20}}$

Explanation:

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

$(\frac{2x^5}{3y^2})^{-4}=(\frac{3y^2}{2x^5})^4$

Apply the exponent within the parentheses and simplify.

$\frac{(3y^2)^4}{(2x^5)^4}$

$\frac{3^4(y^2)^4}{2^4(x^5)^4}$

$\frac{81y^{(2\times 4)}}{16x^{(5\times 4)}}$

$\frac{81y^8}{16x^{20}}$

This fraction cannot be simplified further.

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Example Question #9 : How To Find The Exponent Of Variables

Simplify:

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

Possible Answers:

$\frac{4z^6}{x^6y^2}$

$\frac{4z^8}{x^4y^2}$

$\frac{4z^6}{x^4y^2}$

$\frac{z^6}{x^4y^2}$

$\frac{4z^6}{x^4y}$

Correct answer:

$\frac{4z^6}{x^4y^2}$

Explanation:

First, recognize that raising the fraction to a negative power is the same as raising the inverted fraction to a positive power.

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

Apply the exponent within the parentheses and simplify.

$\frac{(2z^3)^2}{(x^2y)^2}$

$\frac{2^2(z^3)^2}{(x^2)^2y^2}$

$\frac{4z^{(3\times 2)}}{x^{(2\times 2)}y^2}$

$\frac{4z^6}{x^4y^2}$

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Example Question #10 : How To Find The Exponent Of Variables

Simplify if $x\neq 0,-1$ and $y\neq 0$ .

$\frac{x^3y+2x^2y+xy}{xy(x+1)}$

Possible Answers:

2x+1

2x-1

x+1

x-1

Correct answer:

x+1

Explanation:

Begin by factoring the numerator and denominator. can be factored out of each term.

$\frac{x^3y+2x^2y+xy}{xy(x+1)}=\frac{xy(x^2+2x+1)}{xy(x+1)}$

can be canceled, since it appears in both the numerator and denomintor.

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

Next, factor the numerator.

$\frac{(x^2+2x+1)}{x+1}=\frac{(x+1)(x+1)}{x+1}$

Simplify.

$\frac{(x+1)(x+1)}{x+1}=x+1$

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All ISEE Upper Level Math Resources

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ISEE Upper Level Math : Variables and Exponents

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #11 : Variables And Exponents

Example Question #12 : Variables And Exponents

Example Question #11 : Variables And Exponents

Example Question #12 : Variables And Exponents

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

Example Question #13 : Variables And Exponents

Example Question #14 : Variables And Exponents

Example Question #5 : How To Find The Exponent Of Variables

Example Question #9 : How To Find The Exponent Of Variables

Example Question #10 : How To Find The Exponent Of Variables

All ISEE Upper Level Math Resources

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