Math: Basic Number Rules

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Questions and Answers

Identify the natural number 48 as prime or composite. If the number is composite, find its prime factorization.

  • prime
  • 2, 2, 2, 2, 3 (correct)
  • 4 · 8
  • 2, 2, 3, 3

Evaluate the expression. $-3^2$

  • -9 (correct)
  • 9
  • 6
  • -6

Evaluate using the order of operations. $4 \cdot (6+5 \cdot 9 - 1)$

  • -46 (correct)
  • -34
  • -48
  • 54

Evaluate using the order of operations. $(-2)^3 + 13 \cdot 5(4 - 2)$

<p>-5 (D)</p> Signup and view all the answers

Evaluate using the order of operations. $2 - [8(2-4) - 6]$

<p>-44 (A)</p> Signup and view all the answers

Evaluate using the order of operations. $3 \cdot 6(-1)^4 - 25 \div 5$

<p>13 (D)</p> Signup and view all the answers

Evaluate using the order of operations. $\frac{-2 + 3^2 - (-8)}{2 - 9 + 12}$

<p>3 (A)</p> Signup and view all the answers

Evaluate using the order of operations. $-5|-4(11 - 9)|$

<p>-40 (B)</p> Signup and view all the answers

Evaluate the expression $-4x + y$ using the given values $x = 3, y = -2$.

<p>-14 (B)</p> Signup and view all the answers

Evaluate the expression $-6xy + 2y - 9$ using the given values $x = 2, y = 1$.

<p>-19 (A)</p> Signup and view all the answers

Evaluate the expression $5x^3 - 3x^2 + 4x + 2$ for $x = -2$.

<p>-58 (B)</p> Signup and view all the answers

Evaluate the expression $\frac{4x - 6y}{x + 12}$ using $x = -2, y = 3$.

<p>$-\frac{13}{5}$ (C)</p> Signup and view all the answers

Evaluate the expression $\frac{2y - x + 18}{2x - y}$ given $x = -1, y = -7$.

<p>1 (C)</p> Signup and view all the answers

Evaluate the expression $|x - y|$ using $x = -3, y = 8$.

<p>11 (C)</p> Signup and view all the answers

Evaluate the expression $|5x + 8y|$ using $x = -2, y = -3$.

<p>34 (B)</p> Signup and view all the answers

Use the formula $C = \frac{5}{9}(F - 32)$ for converting degrees Fahrenheit into degrees Celsius. Convert $F = 212^\circ$ to Celsius.

<p>$100^\circ C$ (D)</p> Signup and view all the answers

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $\frac{5}{6} - \frac{1}{8}$

<p>$\frac{17}{24}$ (D)</p> Signup and view all the answers

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $(\frac{3}{2}) \cdot (\frac{3}{2}) \cdot (\frac{3}{8})$

<p>$\frac{45}{16}$ (A)</p> Signup and view all the answers

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $\frac{3}{20} \div \frac{7}{12}$

<p>$\frac{9}{35}$ (C)</p> Signup and view all the answers

Perform the operations. $-12.5 - (-11.7)$

<p>-0.8 (A)</p> Signup and view all the answers

Perform the operations. $(6.7 - 8) \div 4 - 2 \cdot \sqrt{0.25}$

<p>-1.325 (B)</p> Signup and view all the answers

List all the elements of $B = {16, \sqrt{8}, -4, 0, \frac{8}{9}, -\frac{9}{8}, 7.5}$ that belong to the set Integers (Z).

<p>{16, -4, 0} (D)</p> Signup and view all the answers

List all the elements of $B = {19, \sqrt{7}, -9, 0, \frac{0}{3}, -3, 4.4}$ that belong to the set Natural numbers (N).

<p>{19} (C)</p> Signup and view all the answers

List all the elements of $B = {14, \sqrt{7}, -17, 0, \frac{0}{7}, 0.47}$ that belong to the set Rational numbers (Q).

<p>{14, -17, 0, $\frac{0}{7}$, 0.47} (B)</p> Signup and view all the answers

List all the elements of $B = {12, \sqrt{8}, -22, 0, \frac{0}{4}, 0.09, -8\pi, 0.252525...}$ that belong to the set Irrational numbers.

<p>{\sqrt{8}, -8$\pi$} (A)</p> Signup and view all the answers

Write the rational number $-\frac{7}{9}$ as a decimal and state whether the decimal is repeating or terminating.

<p>-0.7 repeating (C)</p> Signup and view all the answers

Indicate the property of real numbers. $3 + 4 = 4 + 3$

<p>Commutative property of addition (C)</p> Signup and view all the answers

Indicate the property of real numbers. $14(tu) = (14t)u$

<p>Associative property of multiplication (A)</p> Signup and view all the answers

Indicate the property of real numbers. $8n + 8m = 8(n + m)$

<p>Distributive property (D)</p> Signup and view all the answers

Indicate the property of real numbers. $4 + (-4) = 0$

<p>Additive inverse (D)</p> Signup and view all the answers

Indicate the property of real numbers. $-4\frac{3}{7} + 0 = -4\frac{3}{7}$

<p>Additive identity (D)</p> Signup and view all the answers

Indicate the property of real numbers. $(-3.1)1 = -3.1$

<p>Multiplicative identity (A)</p> Signup and view all the answers

Indicate the property of real numbers. $\frac{1}{5} \cdot 5 = 1$

<p>Multiplicative inverse (A)</p> Signup and view all the answers

Insert <, >, or = to make the statement true. $7 ____ -6$

<blockquote> <p>(A)</p> </blockquote> Signup and view all the answers

Insert <, >, or = to make the statement true. $-6.4 ____ -8.2$

<blockquote> <p>(A)</p> </blockquote> Signup and view all the answers

Insert <, >, or = to make the statement true. $1/3 ____ 0.33$

<blockquote> <p>(B)</p> </blockquote> Signup and view all the answers

Insert <, >, or = to make the statement true. $3.14 ____ \pi$

<p>&lt; (C)</p> Signup and view all the answers

Use the given real number line to compute the distance. Find d(A, B).

<p>9 (B)</p> Signup and view all the answers

Find the distance between $\frac{7}{15}$ and $-\frac{3}{10}$.

<p>$\frac{23}{30}$ (C)</p> Signup and view all the answers

Find the distance between $-3\frac{5}{6}$ and $-6\frac{3}{8}$.

<p>$2\frac{17}{24}$ (D)</p> Signup and view all the answers

Flashcards

Divisible by 2

Last digit is 0, 2, 4, 6, or 8.

Divisible by 3

Sum of the digits is divisible by 3.

Divisible by 5

Last digit is 5 or 0.

Prime Number

Counting number > 1, only factors are 1 and itself.

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Evaluating an Expression

Find the value of the expression by replacing the variable.

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Order of operations

  1. Parentheses; 2. Exponents; 3. Multiplication and Division (left to right); 4. Addition and Subtraction (left to right).
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Least Common Multiple (LCM)

The smallest number that is a multiple of both numbers.

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Rational Number (Q)

All fractions where top and bottom numbers are integers

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Irrational numbers

numbers that cannot be expressed as a fraction, eg, π, √2 etc.

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Real numbers (R)

all numbers that can be written as a decimal.

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Solving Linear equation STEP 2

Combine like terms in either sides

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Solving Linear equation STEP 3

Remove the smaller term of x, using inverse operations

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Solving Linear equation STEP 1

Distribute, if there are any parentheses

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Formula C

For converting degrees Fahrenheit into degrees Celsius

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Solve a Formula for a Specific Variable

all variables are non-zero

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Distance between two real numbers

|a - b| or d = |b - a|

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Multiplicative Inverse Property: a

is 1 but a ≠ 0

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Additive Identity Property

a + 0 = a

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Additive Inverse Property

a + (− a) = 0

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Multiplicative Identity Property

a × 1 = a

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Study Notes

Divisibility Rules

  • A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8
  • A number is divisible by 3 if the sum of its digits is divisible by 3
  • A number is divisible by 5 if its last digit is 5 or 0

Prime Numbers

  • Determined by counting numbers
  • A prime number is greater than 1
  • Prime numbers only have 1 and itself as a factor

Signed Numbers: Addition & Subtraction

  • The sign of the larger number, in absolute value form is maintained
  • Add numbers if they have the same sign
  • Subtract numbers if they have a different sign

Signed Numbers: Multiplication & Division

  • A positive result occurs if the numbers have the same sign
  • A negative result occurs if the numbers have different signs

Order of Operations

  • Parentheses come first
  • Exponents come second
  • Multiplication and Division (from left to right) come third
  • Addition and Subtraction (from left to right) come last

Evaluating Expressions

  • Find the value of the expression when its variable is replaced with a given number

LCM

  • A least common multiple of two numbers is the smallest multiple for both numbers

Real numbers contain decimals

  • Includes fractions in decimal form
  • For example: 0.5, 0.75, 2.35, -0.073, 0.3333, 2.142857

Real Numbers also contain irrational numbers

  • Includes numbers such as Ï€, √2
  • Every real number corresponds to a point on the number line.

Commutative Property

  • Addition: a + b = b + a
  • Multiplication: a × b = b × a

Associative Property

  • Addition: (a + b) + c = a + (b + c)
  • Multiplication: (a × b) × c = a × (b × c)

Distributive Property of Multiplication over Addition

  • Defines that a(b + c) = ab + ac
  • (a+b)c can also be written as ac + bc

Additive Identity Property

  • a + 0 = a

Additive Inverse Property

  • a + (− a) = 0

Multiplicative Identity Property

  • a × 1 = a

Multiplicative Inverse Property

1

  • a × = 1 when a ≠ 0 a

Distance Between Real Numbers

  • d = |a - b| or d = |b - a|

Solving Linear Equations: General Strategy

  • STEP 1: Distribute if there are any parentheses
  • STEP 2: Combine like terms on either side
  • STEP 3: Remove the smaller term of x, using inverse operations
  • STEP 4: Isolate the term of x, using inverse operations
  • STEP 5: Solve for x by dividing both sides by the coefficient of x

Integer Exponent Rules

  • These rules apply when a, b ≠ 0

Zero-Exponent Rule

  • a0 = 1

Product Rule

  • Defines that am â‹… an = am + n

Quotient Rule

am

  • Defines that = am - n an

One-Exponent Rule

  • a1 = a

Power Rule

  • Defines that (am)n = am â‹… n

Negative Exponent Rule

1

  • Defines that a-n = an

Power of a Product

  • (a â‹… b)m = am â‹… bm

Power of a Quotient

am

am bm

Negative Power of a Quotient

a −m bm

b am

Factoring Special Products

  • a2 - b2 = (a - b)(a + b)
  • a3 - b3 = (a - b)(a2 + ab + b2)
  • a3 + b3 = (a + b)(a2 - ab + b2)

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