Problem 55 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ \left(6^{2}-4^{2}\right) \div 5 $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((6^2 - 4^2) \div 5\). Answer: The simplified value of the given expression is 4.

Step 1: Evaluate the exponentiation

We have two exponentiation terms in the expression: \(6^2\) and \(4^2\). We will calculate these before moving on to the next operation. $$ 6^2 = 6 \times 6 = 36 \\ 4^2 = 4 \times 4 = 16 $$ Now the expression becomes: $$ (36 - 16) \div 5 $$

Step 2: Perform subtraction

Next, we will perform the subtraction inside the parentheses: $$ 36 - 16 = 20 $$ Now the expression becomes: $$ 20 \div 5 $$

Step 3: Perform division

Finally, we will perform the division: $$ 20 \div 5 = 4 $$ So, the simplified value of the given expression is 4.

Key Concepts

Exponentiation

When you're faced with expressions like in the exercise \( (6^2 - 4^2) \div 5 \) involving exponentiation or 'raising to a power', it's crucial to tackle those first. Exponentiation is a form of repeated multiplication, where a number, called the base, is multiplied by itself a certain number of times, indicated by the exponent. For instance: \( 6^2 \) means \( 6 \) multiplied by \( 6 \), which equals 36. Similarly, \( 4^2 \) equals 16. Thinking of exponents as shortcuts can be helpful — they're quick ways to express large multiplications.

Always remember to calculate the exponents before moving on to other operations, except when they're inside parentheses with other numbers or operations — those innermost parentheses get priority.

Parentheses in Math

In mathematical expressions, parentheses are used to indicate which operations should be performed first. They are the highest priority in the order of operations, also known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Whenever you see an expression enclosed in parentheses, like in our example \( (36 - 16) \div 5 \), you must simplify what's inside them before dealing with the rest of the equation.

This hierarchy ensures that everyone solves math problems consistently and obtains the same results. In cases with nested parentheses, work from the innermost pair out towards the outer pairs to simplify the expression step by step.

Simplifying Expressions

Simplifying an algebraic expression means to break it down to the simplest form possible, following the order of operations. It involves performing all multiplications, divisions, additions, and subtractions as dictated by the hierarchy of operations. Using the exercise \(20 \div 5\) as an example, we can see that after dealing with parentheses and exponentiation, division is the next step.

Breaking expressions down in this way, step by step, prevents mistakes and ensures that you end up with the correct simplification. The process is akin to following a recipe — you must mix the ingredients in a specific order to get the desired result. Similarly, with mathematical expressions, each step gets you closer to your 'finished dish,' the simplified solution.