Lesson: CSU ELM Prep - 1.4 Exponents & Order of Operation

Lesson Learning Objectives

In this lesson you will learn:

  1. Properties of basic exponents.
  2. How to apply the order of operation to a numerical expression.
  3. How to distinguish between reasonable and unreasonable claims based on data.
  4. How to evaluate the reasonableness of a solution to a problem.
  5. How to express relationships among quantities using variables.
  6. How to solve a real world application involving order of operation.

Exponent Review

The expression \(3^2\) is simplified by multiplying the base of 3 times itself two times. (3)(3) = 9.

The expression \(2^4\) is simplified by multiplying the base of 2 times itself four times. (2)(2)(2)(2) = 16

The expression \( \left(-3\right)^2 \) is simplified by multiplying the base of (-3) times itself two times. (-3)(-3) = 9.

The expression \( \left(-2\right)^3 \) is simplified by multiplying the base of (-2) times itself three times. (-2)(-2)(-2) = -8

Notice \( \left(-3\right)^2 \) has a positive answer, and \( \left(-2\right)^3 \) has a negative answer.
When is the answer positive and when is the answer negative?
Do you see the pattern, can you create a rule?
Think about it before going to the next slide!

Exponent Rule

To find a rule, try other problems such as:
\( \left(-2\right)^4 \), \( \left(-2\right)^5 \), \( \left(-3\right)^3 \), \( \left(-3\right)^4 \)

When is the answer positive, and when is the answer negative?

\( \left(-2\right)^4 \) = + 16
\( \left(-2\right)^5 \) = -32
\( \left(-3\right)^3 \) = -27
\( \left(-3\right)^4 \) = +81

Exponent Rule
Let a and b be positive real numbers:
\( \left( -a \right)^b \) = positive if b is even
\( \left( -a \right)^b \) = negative if b is odd

Example 1:

Simplify \( \left( -5 \right)^2 \) (Note: The negative number is inside the parentheses.)

Solution:
\( \left( -5 \right)^2 \) (Note: The exponent is even.)
\( \left( -5 \right)\left( -5 \right) \)
25 (Note: The answer is positive )

Therefore, the simplified form is 25.

Example 2:

Simplify \(\left(-5\right)^3\)

Solution:
\( \left(-5\right)^3\) (Note: The exponent is odd.)
\( \left( -5 \right) \left( -5 \right) \left( -5 \right) \)
-125 (Note: The answer is negative)

Therefore, the simplified form is -125.

Exponent Rules:

Exponent Rule: (Negative number inside parentheses)
Let a and b be positive real numbers:
\( \left( -a \right)^b \) = positive if b is even
\( \left( -a \right)^b \) = negative if b is odd

Exponent Rule: (Negative number without parentheses.)
If a is a positive real number, then:
\( -a^{odd~or~even~real~number} \) = negative number

Example 3:

Simplify: \( -5^2 \)

Solution:The exponent is on the 5 and not the -5, so write the problem as follows:
-(5)(5)
-25

Therefore, the simplified form is -25.

Example 4:

Simplify \( -5^3 \)

Solution:
\( -5^3 \)
-(5)(5)(5)
-125

Therefore, the simplified form is -125.

Example 5:

Simplifiy \( -2^2 \)

Solution:
The expondent is on the 2 and not the -2, so write the problem as follows:
-(2)(2)
-4

Therefore, the simplified form is -4

Example 6:

\( -2^3 \)

Solution:
\( -2^3 \)
-(2)(2)(2)
-8

Therefore, the simplified form is -8

Properties

Properties
let x be a real number, then:
  1. Any real number (other than zero) or variable to a power of zero equals one. \( x^0 = 1 \)
  2. Any base )real number or variable) to a power of one equals the base \( x^1 = x \)

Examples of property 1:
a) \(6^0=1\)
b) \(5^0=1\)
c) \(76^0=1\)
d) \( \left( xy^2\right)^0=1\)
e) \( \left( 3x + 2y \right)^0 = 1\)
f) \( 3x^0 = 3(1) = 3 \)
g) \( -5x^0 = -5(1) = -5 \)

Examples of property 2:
a) \( 6^1 = 6 \)
b) \( 5^1 = 5 \)
c) \( 76^1 = 76 \) d) \( \left( xy^2\right)^1 = \left(xy^2\right) \)
e) \( \left( 3x + 2y\right)^1 = \left( 3x + 2y\right) \)
f) \( 3x^1 = 3x \)
g) \( -5x^1 = -5x \)

Power of 10 property

Power of 10 property
let x be a real number. Then any multiple of ten can be written as a power of ten: \( 10^x \)

Examples:
\( 10^1 = 10 \)
\( 10^2 = 100 \)
\( 10^3 = 1,000 \)
\( 10^4 = 10,000 \)
\( 10^5 = 100,000 \)
\( 10^6 = 1,000,000 \)

Student Interactive #1:

Simplify \(-3^2 \), \( \left(-3\right)^2 \),\( \left(-3\right)^3 \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #1:

Simplify \(-3^2 \), \( \left(-3\right)^2 \),\( \left(-3\right)^3 \)

The answers are -9,9,-27, respectively.

Solution:
\( -3^2 \) = -(3)(3) = -9 \)
\( \left(-3\right)^2 = (-3)(-3) = 9 \)
\( \left(-3\right)^3 = (-3)(-3)(-3) = -27 \)

Therefore, the answers are -9, 9, -27, respectively.

Student Interactive #2:

Simplify: \( \left( x^2 + y^4 \right)^0 \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #2

Simplify: \( \left( x^2 + y^4 \right)^0 \)

The answer is 1.

Solution:
\( \left( x^2 + y^4 \right)^0 = 1 \)

The exponent zero is on the entire set of parentheses. Thus the value is 1.

Therefore, the simplified form is 1.

Order of Operation

When an expression contains more then one mathematical operation, an order of operation must be applied. When working a problem with more than one operation, execute the operations in the order below:

Order of Operation
1. Parentheses
( ), [ ], { }, absolute value, and the fraction bar.
2. Exponents, Roots (Perform from left to right)
3. Multiplication, Division (Perform from left to right)
4. Addition, Subtraction (Perform from left to right)
  1. Simplify any parentheses, absolute values, and or reduce fractions. (These can all be simplified at the same time.)
  2. simplify exponents and or roots. (These can be simplified at the same time.)
  3. Simplify multiplication and or division.
    If both multiplication and division are present, perform the operations from left to right.
  4. Simplify addition and or subtraction.
    If both addition and subtraction are present, perform the operations from left to right.

Students make a common error when dealing with step 3 and the following form:

Consider: \( 35 \div 4 \cdot 3 \)
This can also be written as \( 36 \div 4 \cdot 3 \)

Incorrect Procedure is:
\( 36 \div \left( 4 \cdot 3 \right) \)
\( 36 \div 12 \)
\( 3 \) WRONG!!!!

Correct Procedures is:
\( \left( 36 \div 4 \right) \cdot 3 \)
\( 9 \cdot 3 \)
\( 27 \) CORRECT!!!!

Since both division and multiplication are on the same level, you must process from left to right. Thus, division must be performed first!

After working with expressions for a while, you will be able to understand how to take shortcuts without violating the rules of order of operation.

Example 7:

Simplify: \( \left(5^2 - 4^2\right)^2 - \left(-50\right)\left(0.2\right) \)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
Since there are mixed operations, use the rules of order of operation to simplify the expression.

\( \left( 5^2 - 4^2 \right)^2 - \left( -50 \right)\left(0.2\right) \)
This expression contains parentheses, exponents, addition/subtraction and multiplication.

Order of operation is:

  1. Parentheses
  2. Exponents
  3. Multiplication
  4. Addition/Subtraction

1) Parentheses:
"Parentheses" means to simplify everything inside the set of parentheses.
\( \left( 5^2 - 4^2 \right)^2 - \left( -50 \right)\left(0.2\right) \)
\( \left( 25 - 16\right)^2 - \left( -50 \right)\left(0.2\right) \)
\( \left( 9 \right)^2 - \left( -50 \right)\left(0.2\right) \)

2) Exponent:
\( \left( 9 \right)^2 - \left( -50 \right)\left(0.2\right) \)
\( 81 - \left( -50 \right)\left(0.2\right) \)

3) Multiplication & Addition
\( 81 - \left( -50 \right)\left(0.2\right) \)
\( 81 - \left( -10 \right) \) The double negative changes to a positive, so the values are added.
\( 81 + 10 \)
\( 91 \)

Therefore, the simplified form is 91.

Example 8:

Simplify: \( 2^0 - 2 -2^2\)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
\( 2^0 - 2 -2^2\) All real numbers, except zero, to the power of zero equals one.
\( 1 - 2 - 4 \)
\( -1 -4 \)
\( -5 \)

Therefore, the simplified form is -5

Example 9:

Simplify: \( 0.4 - \frac{28}{54} \left( \frac{6}{35} \right) \)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
Change 0.4 into a fraction: \( \frac{4}{10} \) Reduce: \( \frac{2}{5} \)
\( \frac{2}{5} - \frac{28}{54}\left( \frac{6}{35} \right) \)

Note: \( \left( \frac{6}{35} \right) \) is just a fraction and does not qualify as being a "parentheses"; it falls under the category of "fraction bar". This expression contains three fraction bars, subtraction, and multiplication. Order of operation is: fraction bars, multiplication, subtraction.

1) Fraction Bars: (The fraction \( \frac{28}{54} \) can be reduced. Factor and reduce. )
\( \require{cancel} \frac{2}{5} - \frac{(\cancel{2})(14)}{(\cancel{2})(27)}\left( \frac{6}{35} \right) \)
\( \frac{2}{5} - \frac{14}{27}\left( \frac{6}{35} \right) \)

2) Multiplication
\( \frac{2}{5} - \frac{14}{27}\left( \frac{6}{35} \right) \) Factor and Reduce
\( \require{cancel} \frac{2}{5} - \left(\frac{2 \cdot \cancel{7}}{\cancel{3} \cdot 9}\right)\left( \frac{2 \cdot \cancel{3}}{\cancel{7} \cdot 5} \right) \)
\( \frac{2}{5} - \frac{4}{45} \)

3) Subtraction:
\( \frac{2}{5} - \frac{4}{45} \) The L.C.D is 45, multiply the first fraction by \( \frac{9}{9} \)
\( \frac{2 \cdot 9}{5 \cdot 9} - \frac{4}{45} \)
\( \frac{18}{45} - \frac{4}{45} \) Subtract
\( \frac{14}{45} \)

Therefore, the simplified form is 14/45.

Example 10:

Simplify: \( \left(-30\right) \div \left(-2\right)\left(-0.4\right)-\left(\frac{-60}{15}\right)-\sqrt{9} - \left( -3 \right)^2 \)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
When a problem contains both fractions and decimals, change the values to either all decimals or all fractions. When performing calculations without a calculator, it is usually easier to work with fractions. Thus, change -0.4 to \( \frac{-4}{10} \) and reduce to \( \frac{-2}{5} \).

The order of operation is:

  1. Fraction bar
  2. Exponents & square root
  3. Multiplication and division from left to right
  4. Subtraction from left to right

\( \left(-30 \right) \div \left(-2\right) \left( - \frac{2}{5} \right) - \left( \frac{-60}{15} \right) - \sqrt{9} - \left( -3 \right)^2 \)
\( \left(-30 \right) \div \left(-2\right) \left( - \frac{2}{5} \right) - \left( -4 \right) - \sqrt{9} - \left( -3 \right)^2 \)
\( \left(-30 \right) \div \left(-2\right) \left( - \frac{2}{5} \right) - \left( -4 \right) - \left(3\right) - 9 \)
\( \left( 15 \right) \left( - \frac{2}{5} \right) - \left( -4 \right) - \left(3\right) - 9 \)
\( \left( - \frac{-30}{5} \right) - \left( -4 \right) - \left(3\right) - 9 \)
\( \left( - 6 \right) - \left( -4 \right) - \left(3\right) - 9 \)
\( \left( - 2 \right) - 3 - 9 \)
\( -5 - 9 \)
\( -14 \)

Therefore, the simplified form is -14.

Student Interactive #3:

Simplify: \( \frac{4}{15} \div \frac{2}{3} \cdot \frac{5}{6} \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #3

Simplify: \( \frac{4}{15} \div \frac{2}{3} \cdot \frac{5}{6} \)

The answer is \( \frac{1}{3} \)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
Order of operation states to perform multiplication and division from left to right. Thus, this problem must be simplified as multiplication and division from left to right.

\( \left( \frac{4}{15} \div \frac{2}{3} \right) \cdot \frac{5}{6} \)
\( \require{cancel} \left( \frac{\cancel{4}}{\cancel{15}} \cdot \frac{\cancel{3}}{\cancel{2}} \right) \cdot \frac{5}{6} \)
\( \require{cancel} \left( \frac{2}{5} \cdot \frac{1}{1} \right) \cdot \frac{5}{6} \)
\( \require{cancel} \frac{\cancel{2}}{\cancel{5}} \cdot \frac{\cancel{5}}{\cancel{6}} \)
\( \frac{1}{3} \)

Therefore, the simplified form is \( \frac{1}{3} \).

Student Interactive #4:

Simplify: \( 0.5 + \frac{5}{3} - 0.6 \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #4:

Simplify: \( 0.5 + \frac{5}{3} - 0.6 \)

The answer is \( \frac{47}{30} \)

Solution:
Change the decimals to fractions: 0.5 = \( \frac{1}{2} \) and 0.6 = \( \frac{6}{10} = \frac{3}{5} \)

\( \frac{1}{2} + \frac{5}{3} - \frac{3}{5} \) The denominators 2, 3 and 5 are all prime numbers, thus the LCD is the product, which is 30.
\( \left( \frac{1}{2} + \frac{5}{3} - \frac{3}{5} \right) = \left( \frac{1 \cdot 15}{2 \cdot 15} + \frac{5 \cdot 10}{3 \cdot 10} - \frac{3 \cdot 6}{5 \cdot 6} \right) \)
\( \left( \frac{15 + 50 - 18}{30} \right) = \frac{47}{30} \) This fraction is reduced to its lowest terms!

Therefore, the simplified form is \( \frac{47}{30} \)

Student Interactive #5

Simplify: \( - \frac{10}{27} \div \frac{15}{18} + 0.375 \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #5

Simplify: \( - \frac{10}{27} \div \frac{15}{18} + 0.375 \)

The answer is \( \frac{-5}{72} \)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
Change 0.375 to a fraction: \( \frac{375}{1000} \) Reduce: \( \frac{3}{8} \)
The order of operation is: fraction bar, division, addition.
This expression contains a fraction bar, division and addition.

1) Fraction Bar: (Only one of the fractions can be reduced )
\( -\frac{10}{27} \div \frac{15}{18} + \frac{3}{8} \)
\( \require{cancel} -\frac{10}{27} \div \frac{(\cancel{3})(5)}{(\cancel{3})(6)} + \frac{3}{8} \)

2) Division:
\( \left( -\frac{10}{27} \div \frac{5}{6} \right) + \frac{3}{8} \)
\( \left( -\frac{10}{27} \cdot \frac{6}{5} \right) + \frac{3}{8} \) Factor and Reduce
\( \require{cancel} \left( -\frac{(2)(\cancel{5})}{{9}{\cancel{3}}} \cdot \frac{(\cancel{3})(2)}{\cancel{5}} \right) + \frac{3}{8} \)
\( -\frac{4}{9} + \frac{3}{8} \)

3) Addition:
\( -\frac{4}{9} + \frac{3}{8} \) Obtain an L.C.D.
The LCD is 72. Multiply the first fraction by \( \frac{8}{8} \), and the second fraction by \( \frac{9}{9} \).
\( -\frac{32}{72} + \frac{27}{72} \)
\( -\frac{5}{72} \)

Therefore, the simplified form is \( \frac{-5}{72} \) .

Student Interactive #6:

Simplify: \( -\frac{6}{21} -\frac{3}{4} \left( \frac{1}{3} \right)^2 \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #6:

Simplify: \( -\frac{6}{21} -\frac{3}{4} \left( \frac{1}{3} \right)^2 \)

The answer is \( \frac{-31}{84} \)

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
This expression contains a fraction bar, subtraction, multiplication and exponents.
The order of operation is: fraction bar, exponents, multiplication, subtraction.

1) Fraction bar:(Only the first fraction can be reduced.)
\( -\frac{6}{21} -\frac{3}{4} \left( \frac{1}{3} \right)^2 \) Factor and Reduce
\( \require{cancel} -\frac{(\cancel{3})(2)}{(\cancel{3})(7)} - \frac{3}{4}\left(\frac{1}{3}\right)^2 \)

2) Exponent:
\( -\frac{2}{7} - \frac{3}{4}\left(\frac{1}{3}\right)^2 \)
\( -\frac{2}{7} - \frac{3}{4}\left(\frac{1}{9}\right) \)

3) Multiplication
\( -\frac{2}{7} - \frac{3}{4}\left(\frac{1}{9}\right) \) Factor and Reduce
\( \require{cancel} -\frac{2}{7} - \frac{\cancel{3}}{4}\left(\frac{1}{(\cancel{3})(3)}\right) \)
\( -\frac{2}{7} - \frac{1}{12} \)
Obtain an L.C.D.
\( -\frac{2}{7} - \frac{1}{12} \) The LCD is 84. Multiply the first fraction by \( \frac{12}{12} \), and the second fraction by \( \frac{7}{7} \).
\( -\frac{2 \cdot 12}{7 \cdot 12} - \frac{1 \cdot 7}{12 \cdot 7} \)

4) Subtraction:
\( - \frac{24}{84} - \frac{7}{84} \)
\( - \frac{31}{84} \)

Therefore, the simplified form is \( \frac{31}{84} \)

Student Interactive #7:

Simplify: \( \left( 0.3 \right)^2 + \left(2\right)^2\left(3\right) - 4.24 + \frac{9}{20} \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #7

Simplify: \( \left( 0.3 \right)^2 + \left(2\right)^2\left(3\right) - 4.24 + \frac{9}{20} \)

The answer is 8.3

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
Since 0.3 squared is easier to manage as a decimal, use long division to change the fraction \( \frac{9}{20} \) into 0.45.

The order of operation is

  1. Exponents
  2. Multiplication
  3. Addition & Subtraction from left to right

\( \left( 0.3 \right)^2 + \left(2\right)^2\left(3\right) - 4.24 + 0.45 \)
\( 0.09 + \left(4\right)\left(3\right) - 4.24 + 0.45 \)
\( 0.09 + 12 - 4.24 + 0.45 \)
\( 12.09 - 4.24 + 0.45 \)
\( 7.85 + 0.45 \)
\( 8.3 \)

Therefore, the simplified form is 8.3.

Student Interactive #8:

Simplify: \( -\left( \frac{-50}{6} \right) + \sqrt{49} - \left(4 -6\right)^2 - \left(-0.3\right)\left(-0.2\right) \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #8

Simplify: \( -\left( \frac{-50}{6} \right) + \sqrt{49} - \left(4 -6\right)^2 - \left(-0.3\right)\left(-0.2\right) \)

The answer is \( \frac{1691}{150} \).

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
Using long division, the fraction \( \frac{50}{6} \) equals a repeating decimal. Therefore, change the decimals into fractions.

\( -\left( \frac{-50}{6} \right) + \sqrt{49} - \left(4 - 6\right)^2 - \left( -\frac{3}{10}\right)\left(-\frac{2}{10}\right) \)
\( -\left( \frac{-25}{3} \right) + \sqrt{49} - \left(-2\right)^2 - \left( -\frac{3}{10}\right)\left(-\frac{2}{10}\right) \)
\( -\left( \frac{-25}{3} \right) + 7 - 4 - \left( -\frac{3}{10}\right)\left(-\frac{2}{10}\right) \)
\( -\left( \frac{-25}{3} \right) + 7 - 4 - \left( \frac{6}{100} \right) \) Reduce
\( -\left( \frac{-25}{3} \right) + 3 - \left( \frac{3}{50} \right) \) The LCD is 150.
\( \left( \frac{25 \cdot 50}{3 \cdot 50} \right) + 3 \left( \frac{150}{150} \right) - \left( \frac{3 \cdot 3}{50 \cdot 3} \right) \)
\( \left( \frac{1250}{150} \right) + \left( \frac{450}{150} \right) - \left( \frac{9}{150} \right) \)
\( \frac{1250 + 450 -9}{150} = \frac{1691}{150} \)

Therefore, the simplified form is \( \frac{1691}{150} \).

Student Interactive #9

Simplify: \( \frac{-3\left[\left(-2\right)^3-\left(-2\right)^2\left(-1\right)\right]}{\left(3-5\right)^2} \)

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #9

Simplify: \( \frac{-3\left[\left(-2\right)^3-\left(-2\right)^2\left(-1\right)\right]}{\left(3-5\right)^2} \)

The answer is 3.

Order of Operation
1. Parentheses, AV, fraction bar.
2. Exponents, Roots
3. Multiplication, Division (left to right)
4. Addition, Subtraction (left to right)

Solution:
\( \frac{-3\left[\left(-2\right)^3-\left(-2\right)^2\left(-1\right)\right]}{\left(3-5\right)^2} \)
\( \frac{-3\left[\left(-8\right)-\left(4\right)\left(-1\right)\right]}{\left(-2\right)^2} \)
\( \frac{-3\left[\left(-8\right)-\left(-4\right)\right]}{\left(-2\right)^2} \)
\( \frac{-3\left[\left(-8\right)+4\right]}{\left(-2\right)^2} \)
\( \frac{-3\left[-8+4\right]}{\left(-2\right)^2} \)
\( \frac{-3\left[-4\right]}{\left(-2\right)^2} \)
\( \frac{-3\left[-4\right]}{4} \)
\( \frac{12}{4} \)
\( 3 \)

Therefore, the simplified form is 3.

Example 11:

A carpenter is framing a picture with wood trim, and the length of the frame is \( 3\frac{2}{3} \) inches.
If the width of the frame is \( 2\frac{1}{4} \) inches, how many inches of the wood is needed to frame the picture?

Solution:
The number of inches needed is the distance around the frame, which is the perimeter of a rectangle.

Change the mixed numbers to improper fractions:
\( 3\frac{2}{3} = \frac{11}{3} \)
\( 2\frac{1}{4} = \frac{9}{4} \)

The preimeter of a rectangle is P = 2L + 2w were L is the length of the rectangle and w is the width.

\( p = 2\left( \frac{11}{3} \right) + 2\left( \frac{9}{4} \right) \)
\( p = \left(\frac{22}{3}\right) + \left( \frac{18}{4} \right) \)
\( p = \left( \frac{22 \cdot 4}{3 \cdot 4} \right) + \left( \frac{18 \cdot 3}{4 \cdot 3} \right) \)
\( p = \left( \frac{88}{12} \right) + \left( \frac{54}{12} \right) \)
\( p = \frac{142}{12} = \frac{71}{6} = 11\frac{5}{6} \)

Therefore, the number of inches of wood needed to frame the picture is: \( 11\frac{5}{6} \)

Example 12:

A local carpet company has been hired to carpet a theater which is in the shape of a trapezoid. The two bases of the trapezoid are 20 yards and 30 yards. If the height of the trapezoid is 25 yards and the cost of the carpet is $10 per yard, find the total cost to carpet the theater.

The area of a trapezoid is given by \( A = \frac{1}{2}h\left(b_1 + b_2\right) \) where \(b_1\) & \( b_2 \) are the bases and h is the height.

The Cost C of the carpet is given by: \( C = \$(10)\frac{1}{2}(h)(b_1+b_2) \overset{{\text{Simplified}}}{\rightarrow} C = \$(5)(h)(b_1+b_2) \)

The total cost to carpet the theater:
\( C = \$(5)(h)(b_1+b_2) \)
\( C = \$(5)(25)(20+30) \)
\( C = \$(5)(25)(50) \)
\( C = \$6250 \)

Therefore, the total cost to carpet the theater is $6250.

Student Interactive #10

If Jeremy and Sam each eat one fifth of a pizza, and Betty eats one fifteenth, how much pizza is left?

Work out the problem with pencil and paper, and then go to the next slide for the solution.

Student Interactive Solution #10

If Jeremy and Sam each eat one fifth of a pizza, and Betty eats one fifteenth, how much pizza is left?

The answer is \( \frac{8}{15} \) of a pizza.

Solution
Find the total amount of pizza consumed, and then determine the remaining amount.

Find the amount of pizza consumed:
\( \text{Amount}_\text{Jeremy} + \text{amount}_\text{Sam} + \text{Amount}_\text{Betty} \): \( \frac{1}{5} + \frac{1}{5} + \frac{1}{15} \rightarrow \text{Find the sum of the 1st two fractions} \rightarrow \frac{2}{5} + \frac{1}{15} \)
\( \frac{2}{5} + \frac{1}{15} = \left( \frac{2 \cdot 3}{5 \cdot 3} + \frac{1}{15}\right) = \left(\frac{6 +1}{15}\right) = \frac{7}{15} \) Thus, \( \frac{7}{15} \) of the pizza has been eaten.

Determine the remaining amount of pizza
One whole pizza - \( \frac{7}{15} \) = ?
\(1 - \frac{78}{15} = \left( \frac{1 \cdot 15}{1 \cdot 15} - \frac{7}{15}\right) = \left( \frac{15-7}{15} \right) = \frac{8}{15} \) Check your answer: Verify by making sure that \( \frac{7}{15} + \frac{8}{15} \) equals one, which represents the whole pizza.

Therefore, the remaining amount is \( \frac{8}{15} \)ths of a pizza