Problem 65 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$24 \div \frac{3^{2}}{8-5}-(-6)$$

Step-by-Step Solution

Answer

The simplified expression equals 14

Step 1: Simplify exponent

First, simplify the exponent: \(3^2\) which is \(3*3=9\). The expression becomes \(24 \div \frac{9}{8-5} - (-6)\)

Step 2: Perform operations in brackets

Next, perform subtraction inside the brackets: \(8 - 5 = 3\). The expression now becomes \(24 \div \frac{9}{3}- (-6)\)

Step 3: Perform division inside the fraction

Perform the division inside the fraction: \(9 \div 3 = 3\). Now the expression looks as follows: \(24 \div 3 - (-6)\)

Step 4: Perform divisions

Now, perform the division: \(24 \div 3 = 8\). The expression, after simplification, becomes \(8 - (-6)\)

Step 5: Perform final subtraction

Finally, perform subtraction: \(8 - (-6) = 8 + 6 =14\)

Key Concepts

Order of Operations

Understanding the order in which mathematical operations should be performed is crucial for simplifying expressions and solving equations accurately. In the mathematical world, this specific sequence is commonly known as PEMDAS or BODMAS.

PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Sometimes, BODMAS is used interchangeably, where 'Brackets' replace 'Parentheses', and 'Orders' stand for 'Exponents'.

Applying Order of Operations in Practice

• Begin by solving any calculations inside parentheses or brackets first.
• Next, address the exponents or any 'orders' in the expression.
• After that, carry out any multiplications and divisions as they appear from left to right.
• Finally, tackle the additions and subtractions, also as they appear from left to right.

Using the correct order ensures the accuracy of the solution and prevents common algebraic errors.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, which is the exponent. This action effectively multiplies the base by itself as many times as the value of the exponent suggests.

For example, if we take the base 3 and raise it to the exponent 2 (written as \(3^2\)), it means we are multiplying 3 by itself once, ending up with \(3 \times 3 = 9\).

Key Points to Remember in Exponentiation

• The expression \(a^n\) tells you to multiply the base 'a' by itself 'n' times.
• If the exponent is 1, the result is always the base itself (\(a^1 = a\)).
• Zero as an exponent results in one (\(a^0 = 1\)), regardless of the base value, except when the base is also zero.
• Negative exponents represent the reciprocal of the base raised to the positive exponent (\(a^{-n} = \frac{1}{a^n}\)).

When simplifying expressions with exponents, calculate these first after any parentheses, as per the order of operations.

Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations form the building blocks of mathematics, and mastering them is essential for progressing into more complex areas.

In performing these operations, certain rules and properties help to streamline calculations:

Properties of Arithmetic Operations

• Commutative property: For addition and multiplication, changing the order of the numbers does not affect the result (e.g., \(2 + 3 = 3 + 2\), \(4 \times 5 = 5 \times 4\)).
• Associative property: When adding or multiplying, the way you group the numbers (using parentheses) does not change the result (e.g., \((2 + 3) + 4 = 2 + (3 + 4)\), \((2 \times 3) \times 4 = 2 \times (3 \times 4)\)).
• Distributive property: Multiplication distributes over addition (e.g., \(a \times (b + c) = (a \times b) + (a \times c)\)).

Understanding these properties within the context of the order of operations will greatly enhance the ability to simplify mathematical expressions effectively and accurately.