# 2.9 Proficiency exam

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.This module contains the proficiency for the chapter "Basic Properties of Real Numbers".

## Proficiency exam

For the following problems, simplify each of the expressions.

( [link] ) $8\left(6-3\right)-5\cdot 4+3\left(8\right)\left(2\right)÷4\cdot 3$

40

( [link] ) ${\left\{2{\left(1+7\right)}^{2}\right\}}^{0}$

1

( [link] ) $\frac{{1}^{8}+{4}^{0}+{3}^{3}\left(1+4\right)}{{2}^{2}\left(2+15\right)}$

$\frac{137}{68}$

( [link] ) $\frac{2\cdot {3}^{4}-{10}^{2}}{4-3}+\frac{5\left({2}^{2}+{3}^{2}\right)}{11-6}$

75

( [link] ) Write the appropriate relation symbol $\left(>,<\right)$ in place of the $*$ .
$5\left(2+11\right)\ast 2\left(8-3\right)-2$

$>$

For the following problems, use algebraic notation.

( [link] ) $\left(x-1\right)$ times $\left(3x\text{\hspace{0.17em}}\text{plus}\text{\hspace{0.17em}}2\right)$ .

$\left(x-1\right)\left(3x+2\right)$

( [link] ) A number divided by twelve is less than or equal to the same number plus four.

$\frac{x}{12}\le \left(x+4\right)$

( [link] ) Locate the approximate position of $-1.6$ on the number line.

( [link] ) Is 0 a positive number, a negative number, neither, or both?

Zero is neither positive nor negative.

( [link] ) Draw a portion of the number line and place points at all even integers strictly between 14 and 20.

( [link] ) Draw a portion of the number line and place points at all real numbers strictly greater than $-1$ but less than or equal to 4.

( [link] ) What whole numbers can replace $x$ so that the following statement is true? $-4\le x\le 5$ .

$0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4,\text{\hspace{0.17em}}5$

( [link] ) Is there a largest real number between and including 6 and 10? If so, what is it?

yes; 10

( [link] ) Use the commutative property of multiplication to write $m\left(a+3\right)$ in an equivalent form.

$\left(a+3\right)m$

( [link] ) Use the commutative properties to simplify $3a4b8cd$ .

$96abcd$

( [link] ) Use the commutative properties to simplify $4\left(x-9\right)2y\left(x-9\right)3y$ .

$24{y}^{2}{\left(x-9\right)}^{2}$

( [link] ) Simplify 4 squared times $x$ cubed times $y$ to the fifth.

$16{x}^{3}{y}^{5}$

( [link] ) Simplify $\left(3\right)\left(3\right)\left(3\right)aabbbbabba\left(3\right)a$ .

$81{a}^{5}{b}^{6}$

For the following problems, use the rules of exponents to simplify each of the expressions.

( [link] , [link] ) ${\left(3a{b}^{2}\right)}^{2}{\left(2{a}^{3}b\right)}^{3}$

$72{a}^{11}{b}^{7}$

( [link] , [link] ) $\frac{{x}^{10}{y}^{12}}{{x}^{2}{y}^{5}}$

${x}^{8}{y}^{7}$

( [link] , [link] ) $\frac{52{x}^{7}{y}^{10}{\left(y-{x}^{4}\right)}^{12}{\left(y+x\right)}^{5}}{4{y}^{6}{\left(y-{x}^{4}\right)}^{10}\left(y+x\right)}$

$13{x}^{7}{y}^{4}{\left(y-{x}^{4}\right)}^{2}{\left(y+x\right)}^{4}$

( [link] , [link] ) ${\left({x}^{n}{y}^{3m}{z}^{2p}\right)}^{4}$

${x}^{4n}{y}^{12m}{z}^{8p}$

$\frac{{\left(5x+4\right)}^{0}}{{\left(3{x}^{2}-1\right)}^{0}}$

1

$\frac{{x}^{\nabla }{x}^{\square }{y}^{\Delta }}{{x}^{\Delta }{y}^{\nabla }}$

( [link] , [link] ) What word is used to describe the letter or symbol that represents an unspecified member of a particular collection of two or more numbers that are clearly defined?

a variable