1. Fundamental Concepts of Algebra

Simplify Exponential Expressions

Exponents and Scientific Notation

Simplifying expressions using the Laws of Exponents

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27

In Exercises 29–34, find the union of the sets. {e,m,p,t,y} ∪ ∅

In Exercises 83–90, evaluate each expression without using a calculator. 27^(1/3)

Determine whether each statement is true or false. {x | x is a natural number less than 3}= {1, 2}

Understanding Exponents and Their Operations

Use the quotient rule to simplify the expressions in Exercises 45-46. √(121/4)

Write each fraction in lowest terms. 18/90

Multiply or divide as indicated. 12.8 * 9.1

Insert ∈ or ∉ in each blank to make the resulting statement true. 5 ____ {4, 6, 8, 10}

Determine whether each statement is true or false. ∅ ∪ ∅ = ∅

Simplifying Radical Expressions Involving Variables - Example 1

In Exercises 65–76, write each number in decimal notation without the use of exponents. −3.14×10^−3

Evaluate each expression. 15/5*4/6-8 / -6-(-5)-8/2

Identify the property illustrated in each statement. Assume all variables represent real numbers. 6*12+6*15=6(12+15)

Write each percent as a fraction. Give answers in lowest terms. 2%

In Exercises 61–66, evaluate each algebraic expression for x=2 and y=-5. |x|+|y|

Determine whether each statement is true or false. ∅ ∩ ∅ = ∅

In Exercises 77–86, write each number in scientific notation. 0.0083

In Exercises 21–28, find the intersection of the sets. {s,e,t}∩{t,e,s}

In Exercises 65–76, write each number in decimal notation without the use of exponents. −8.17×10^6

Simplifying Expressions with Negative Exponents - Ex 1

Multiply or divide as indicated. Write answers in lowest terms as needed. (4/5)*(6/7)

Write each improper fraction as a mixed number. 17/12

For the decimal number 46.249, round to the place value indicated. 
(a) hundredths (b) tenths (c) ones or units (d) tens

In Exercises 77–86, write each number in scientific notation. −3829

In Exercises 87–106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. (6.1×10^−8)(2×10^−4)

Exponent Rules: The Power to Power Rule!

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. (M ∩ N) ∪ R

Simplify each exponential expression: (-5x^3y^2)(-2x^(-11)y^(-2))

Evaluate each expression. (5-3²)(√16-2³)

Complete the table of fraction, decimal and percent equivalents. 
Fraction in lowest terms(or Whole Number) ?
Decimal 0.05
Percent 5%

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. {x | x ∈ M and x ∈ Q}

Add or subtract as indicated. Write answers in lowest terms as needed. 7/12 - 1/3

Complete the table of fraction, decimal and percent equivalents.
Fraction in lowest terms(or Whole Number) ?
Decimal ?
Percent 50%

Write a decimal number that has 5 in the thousands place, 0 in the tenths place, and 4 in the ten-thousandths place.

Applying the Rules of Exponents - Basic Examples #1

Find each sum or difference. 9/10 - (-4/3)

Determine whether each statement is true or false. |25| = |-25|

Find each product or quotient where possible. 5(-7)

In Exercises 35–38, list all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. {-9, -4/5, 0, 0.25, √3, 9.2, √100}

Write each improper fraction as a mixed number. 16/9

In Exercises 51–60, rewrite each expression without absolute value bars. ||-3|-|-7||

State the name of the property illustrated: 3+17 = 17+3

Power of a Quotient Property

Power of a Product Property

Determine whether each statement is true or false. |11| * |-6| = |-66|

Add or subtract as indicated. Write answers in lowest terms as needed. 4(3/4) - 1(2/5)

Insert ⊆ or s in each blank to make the resulting statement true. {5, 6, 7, 8} ____ {1, 2, 3, 4, 5, 6, 7}

Add or subtract as indicated. 46.88 - 13.45

Determine whether each statement is true or false. 1 ∈ {11, 5, 4, 3, 1}

Multiply or divide as indicated. Write answers in lowest terms as needed. 3(1/4)*1(2/3)

Use set notation, and list all the elements of each set. {74, 68, 62, ..., 38}

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. N′

Write each decimal as a fraction. (Do not write in lowest terms.) 0.104

Determine whether each statement is true or false. 9 ∉ {8, 5, 2, 1}

Perform the indicated operation and write the answer in decimal notation. (3*10^3)(1.3*10^2)

In Exercises 51–60, rewrite each expression without absolute value bars. |12 - π|

Simplify an Algebraic Expression by Combining Like Terms

Let A = { -6, - 12/4 , - 5/8 , - √3, 0, 1/4 , 1, 2π, 3, √12}. List all the elements of A that belong to each set. Irrational numbers

The mixed number equivalent of the improper fraction 31/5 is 6(1/5).

Simplify expressions with 0 and negative exponents

Round each decimal to the nearest thousandth.
(a) 0.8 (line above 8) (b) 0.4 (line above 4) (c) 0.9762 (d) 0.8645

Evaluate each expression. [-8+(-4)(-6)/12] / [4-(-3)]

Find {16, 18, 21, 50} ∪ {15, 16, 17, 18}.

Identify the property illustrated in each statement. Assume all variables represent real numbers. 5(t+3) = (t+3)*5

In Exercises 67–74, express the distance between the given numbers using absolute value. Then find the distance by evaluating the absolute value expression. 2 and 17

Multiply or divide as indicated. 34.04 * 0.56

Simplify each exponential expression in Exercises 23–64. (−3x^2 y^5)^2

Simplify each exponential expression: (-2x^4y^3)^3

Negative Exponents

Write each number in decimal notation: 7.45*10^(-5)

Multiply numbers expressed in scientific notation

Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. A ⊆ U

Write each percent as a fraction. Give answers in lowest terms. 180%

Write each fraction in lowest terms. 16/20

In Exercises 51–60, rewrite each expression without absolute value bars. |7 - π|

Let U = {1, 2, 3, 4, 5} and A = {1, 2, 3}. Find A′.

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. ∅ ∩ Q

Power of a Quotient Rule

Multiply or divide as indicated. 124.03 /100

Add or subtract as indicated. Write answers in lowest terms as needed. 3(1/8) + 2(1/4)

Complete the table of fraction, decimal and percent equivalents.
Fraction in lowest terms(or Whole Number) 1/50
Decimal ?
Percent 2%

Applying the Rules of Exponents - Basic Examples #3

In Exercises 75–84, state the name of the property illustrated. 6+(2+7)=(6+2)+7

Identify each set as finite or infinite. Then determine whether 10 is an element of the set. {4, 5, 6,...., 15}

Let A = {a, b, c}, B = {a, c, d, e}, and C = {a, d, f, g}. Find the indicated set A ∩ B.

Identify the property illustrated in each statement. Assume all variables represent real numbers. (7.5-y)+0=7.5-y

Simplify each exponential expression in Exercises 23–64. (−5x^4 y)(−6x^7 y^11)

Complete the table of fraction, decimal and percent equivalents.
Fraction in lowest terms(or Whole Number) 1/10
Decimal ?
Percent ?

Multiply or divide as indicated. Write answers in lowest terms as needed. (3/20)*(5/21)

GED Sample Test with Solutions , Question #42

Determine whether each statement is true or false. {5, 7, 9, 19} ∩ {7, 9, 11, 15} = {7, 9}

Multiply or divide as indicated. Write answers in lowest terms as needed. (3/4)/12

Add or subtract as indicated. Write answers in lowest terms as needed. 7/12-1/9

Determine whether each statement is true or false. |-14| / |2| = |-14/2|

Add or subtract as indicated. 3.87 + 15 + 2.9

Write each fraction in lowest terms. 8/16

Divide exponential expressions (part 2)

In Exercises 35–38, list all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. {-11, -5/6, 0, 0.75, √5, π, √64}

In Exercises 87–106, perform the indicated computations. Write the answers in scientific notation. If necessary, round the decimal factor in your scientific notation answer to two decimal places. (2.4×10^−2)/(4.8×10^-6)

In Exercises 77–86, write each number in scientific notation. 579,000,000,000,000,000

Multiply or divide as indicated.19.967/9.74

Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. A ⊆ C

In the decimal number 367.9412, name the digit that has each place value. 
(a) tens (b) tenths (c) thousandths (d) ones or units (e) hundredths

Insert ∈ or ∉ in each blank to make the resulting statement true. 0 _____ {0, 5, 6, 7, 8, 10}

Add or subtract as indicated. Write answers in lowest terms as needed. 7/9-2/9

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √125x^2

Simplify exponential multiplication expressions

Multiply or divide as indicated. 57.116 * 100

Write each fraction in lowest terms. 90/150

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. {x | x ∈ U, x ∉ R}

Write each decimal as a fraction. (Do not write in lowest terms.) 0.043

Determine whether each statement is true or false. {2, 5, 8, 9} = {2, 5, 9, 8}

Divide exponential expressions (part 1)

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. M ∩ N

Fill in the blank(s) to correctly complete each sentence. In the expression 6^3, 6 is the ____ , and 3 is the ____ .

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. M ∪ N

Simplifying Expressions with Negative Exponents - Ex 3

Add or subtract as indicated. 25.32 + 109.2 + 8.574

In Exercises 75–84, state the name of the property illustrated. 7⋅(11⋅8)=(11⋅8)⋅7

Simplify using properties of exponents. [5x^(2/3)][4x^(1/4)]

Evaluate each exponential expression: (3^3)/(3^6)

Find each product or quotient where possible. -8(-5)

Use set notation, and list all the elements of each set. {x | x is a natural number not greater than 4}

Write each decimal as a fraction. (Do not write in lowest terms.) 0.6

In Exercises 75–84, state the name of the property illustrated. 1/(x+3) (x+3)=1, x≠−3

Write each number in scientific notation. 3,590,000

Write each percent as a fraction. Give answers in lowest terms. 7.5%

Intro to Evaluating Exponents - A few True/False Questions - Ex1

Multiply or divide as indicated. 44.4788/5.27

Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. C ⊆ U

Power of a Product Rule

In Exercises 61–66, evaluate each algebraic expression for x=2 and y=-5. |x+y|

Divide numbers in scientific notation

Determine whether each statement is true or false. 6 ∈ {2, 5, 8, 9}

Multiply or divide as indicated. Write answers in lowest terms as needed. (24/7)/(6/21)

Raise an exponential expression to a power

Find each product or quotient where possible. -10/17 / (-12/5)

Use set notation and write the elements belonging to the set 
{x | x is a natural number less than 6}.

Evaluate each exponential expression: (-3)^3 (-2)^2

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. Q ∩ R′

Multiply or divide as indicated. Write answers in lowest terms as needed. (6/11)/(5/4)

Determine whether each statement in Exercises 43–50 is true or false. -π ≥ - π

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8},
N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. Q ∩ (M ∪ N)

Evaluate each exponential expression: 2^(-4) + 4^(-1)

State the name of the property illustrated: (6 • 3) • 9 = 6 • (3 • 9)

Find each product or quotient where possible. 0/-8

Let A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, andU = {2, 4, 6, 8, 10, 12, 14}. Determine whether each statement is true or false. B ⊆ C

Use set notation, and list all the elements of each set. {12, 13, 14,..., 20}

Write each percent as a fraction. Give answers in lowest terms. 15%

Multiply or divide as indicated. 78.65/11

Use the product rule to simplify the expressions in Exercises 41 - 44. In exercises 43 - 44, assume that variables represent nonnegative real numbers. √300

Write each percent as a fraction. Give answers in lowest terms. 47%

Identifying Like Terms

Evaluate each expression. (-2/9 -1/4) - {-5/18 - (-1/2)}

Identify the set { 1,1/3, 1/9 ,1/27, ....} as finite or infinite.

Write each fraction as a decimal. For repeating decimals, write the answer by first using bar notation and then rounding to the nearest thousandth. 5/9

Find each product or quotient where possible. -0.5(-0.6)

In Exercises 91–100, simplify using properties of exponents. (20x^1/2)/(5x^1/4)

Insert ⊆ or s in each blank to make the resulting statement true. ∅ ____ {1, 4, 6, 8}

In the fraction 5/8, 5 is the numerator and 8 is the denominator.

Insert ⊆ or s in each blank to make the resulting statement true. {1, 5} ____ {0, 1, 2, 3, 5}

Insert ∈ or ∉ in each blank to make the resulting statement true. {0} _____ {0, 1, 2, 5}

Multiply or divide as indicated. Write answers in lowest terms as needed. 6(3/4)/(3/8)

Write each percent as a fraction. Give answers in lowest terms. 35%

Determine whether each statement is true or false. {3, 5, 9, 10} ∩ ∅ = {3, 5, 9, 10}

Insert ⊆ or s in each blank to make the resulting statement true. ∅ ____ ∅

The formula C=5/9(F-32) expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. In Exercises 17–18, use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale. 50 °F

Multiply or divide as indicated. Write answers in lowest terms as needed. (7/5)/(3/10)

Insert ⊆ or s in each blank to make the resulting statement true. {2, 4, 6} ____ {2, 3, 4, 5, 6}

Zero Exponent Rule

Write each improper fraction as a mixed number. 77/12

Evaluate each exponential expression: 5^(-3) • 5

Use the distributive property to calculate each value mentally. 72*17+28*17

Find each product or quotient where possible. 7/6 / -2/3

Use the distributive property to calculate each value mentally. 123 5/8 * 1 1/2 - 23 5/8 * 1 1/2

Simplifying Expressions with Negative Exponents - Ex 2

Complete the table of fraction, decimal and percent equivalents. 
Fraction in lowest terms(or Whole Number) 1/3
Decimal ?
Percent ?

Determine whether each statement is true or false. 3 ∈ {2, 5, 6, 8}

Evaluate each expression. (4-2³)(-2+√25)

Complete the table of fraction, decimal and percent equivalents.
Fraction in lowest terms(or Whole Number) 1
Decimal 1.0
Percent ?

Find each product or quotient where possible. 100/-25

Write each percent as a fraction. Give answers in lowest terms. 2.5%

Complete the table of fraction, decimal and percent equivalents. 
Fraction in lowest terms(or Whole Number) ?
Decimal ?
Percent 20%

Use set notation, and list all the elements of each set. {17, 22, 27, .. , 47}

Multiply or divide as indicated. 22.41 * 33

Multiply or divide as indicated. Write answers in lowest terms as needed. 8/(4/9)

Multiply or divide as indicated. Write answers in lowest terms as needed. 2(1/2)/1(5/7)

Write each mixed number as an improper fraction. 2(3/5)

Find each product or quotient where possible. -14/2

Add or subtract as indicated. 28.73 - 3.12

Use set notation, and list all the elements of each set. {x | x is a natural number greater than 8 and less than 15}

Add or subtract as indicated. Write answers in lowest terms as needed. 13/15 - 3/15

Let A = {a, b, c}, B = {a, c, d, e}, and C = {a, d, f, g}. Find the indicated set A ∪ B.

In Exercises 21–28, find the intersection of the sets. {a,b,c,d}∩∅

Multiply or divide as indicated. 0.094 * 1000

Find each product or quotient where possible. 5/0