Problem 30 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\left(4^{2}-2 \cdot 4\right)-2^{3}\)

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Calculate the Exponents

Firstly, calculate the exponent parts of the expression: \[4^2 = 16 \] and \[2^3 = 8\] This will simplify the expression to:\[(16 - 2 \cdot 4) - 8 \].

Step 2: Perform Multiplication

Next, perform the multiplication operation within the parentheses: \[2 \cdot 4 = 8 \]. This changes the expression to: \[(16 - 8) - 8\].

Step 3: Evaluate the Parentheses

Now, resolve the expression inside the parentheses by performing the subtraction:\[16 - 8 = 8\].The expression is now simplified to:\[8 - 8\].

Step 4: Perform Final Subtraction

Finally, subtract the remaining values:\[8 - 8 = 0\].This is the value of the entire expression.

Key Concepts

Exponents

Exponents are a fundamental part of mathematics. They tell us how many times to multiply a number by itself. For example, in the expression \(4^2\), the number 4 is the base and 2 is the exponent. This means you multiply 4 by itself once:
\(4^2 = 4 \times 4 = 16\).
Exponents are essential in simplifying complex mathematical expressions, as they allow for compact representation of repeated multiplication.
When dealing with exponents in mathematical operations, always process them first before other operations except inside parentheses. This is a crucial part of the order of operations, following the rules of PEMDAS/BODMAS.

Multiplication

Multiplication is one of the four primary arithmetic operations. It involves calculating the product of numbers. In the expression, after computing the exponents, you need to handle the multiplication:
\(2 \cdot 4\).
This refers to calculating twice the value of 4, resulting in \(8\).
Multiplication follows exponents in the order of operations unless it's inside parentheses. It is often necessary to simplify expressions into digestible parts, which makes solving them easier.

Parentheses

The use of parentheses in mathematical expressions signifies that operations within them should be performed first. In our exercise, after calculating multiplication, we need to evaluate the expression within parentheses:
\((16 - 8)\).
By performing the subtraction within the parentheses, we get \(8\).
Parentheses are vital for altering the normal order of operations, especially when complex expressions are involved. They help ensure that calculations are done correctly by specifying which operations should be completed first. This makes solving problems more structured and precise.

Subtraction

Subtraction is the operation of finding the difference between numbers. Once all other operations such as exponents, multiplication, and parentheses are resolved, subtraction is performed.
In the exercise, after simplifying the expression via parentheses, the final step involves subtracting:
\(8 - 8\) which gives \(0\).
This simple arithmetic operation follows at the end of our order of operations. Proper execution of subtraction, after resolving all other components, ensures accuracy in arriving at the final answer.