Problem 14 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$y^{4} \div 8 \text { when } y=4$$

Step-by-Step Solution

Answer

The expression evaluates to 32 when y = 4.

Step 1: Substitute the given value

Substitute y = 4 into the given expression, so it becomes \(4^{4} \div 8 \).

Step 2: Apply exponent

Calculate \(4^{4}\) first, according to the order of operations(BIDMAS/BODMAS) which says that exponentiation should be performed before division. \(4^{4}\) equals 256.

Step 3: Perform division

Now divide the result of \(4^{4}\) by 8. That is \(256 \div 8\) equals 32. Hence the expression evaluates to 32 when y = 4.

Key Concepts

Order of Operations

Math can be tricky because there are rules about the order in which operations must be performed. This is often remembered by the acronym BIDMAS/BODMAS, which stands for Brackets, Indices (Exponents), Division and Multiplication, Addition and Subtraction. Following these rules ensures that everyone calculates the same way and gets the same answer.
In any math expression, we first solve anything inside brackets, then handle exponents or indices next.

• After that, do any division or multiplication as they appear, working left to right.
• Finally, tackle addition and subtraction, also from left to right.

In the exercise example, we follow the order of operations by handling the exponentiation before division. Specifically, we calculated \(4^{4}\) before dividing by 8. Ignoring this order might lead to a wrong answer. Understanding and practicing this concept is key to mastering math problems.

Substitution Method

The substitution method involves replacing a variable in an expression with a given value. It's like plugging in numbers where the letters are. This method helps us solve or simplify expressions by making them more concrete. When we know what value a variable stands for, we can substitute it to perform calculations with real numbers.

• In our exercise, we replaced \(y\) with 4 in the expression \(y^{4} \div 8\).
• This turned the expression into \(4^{4} \div 8\), making it ready for calculation.

Substitution is often the first step in solving algebraic problems. By doing this, we reduce the expression to simpler numerical calculations. This simplification makes it easier to apply the order of operations, as seen in our example.

Division

Division is the process of determining how many times one number is contained within another. It's one of the four basic operations in math. In the context of our exercise, division was the final step after simplifying the expression using substitution and exponentiation.

• We started with \(4^{4} = 256\). This meant we had a resulting number from our exponentiation.
• We then divided 256 by 8 to complete the calculation.

Performing division correctly is crucial to getting the right answer. It involves distributing the total number (256 in our case) equally into a certain number of parts (8 here). The division yields 32, which is our answer.