Problem 17 - Fundamentals of Mathematics

Question

Determine the value of each of the following. \(4\left(6^{2}-3^{3}\right) \div\left(4^{2}-4\right)\)

Step-by-Step Solution

Answer

The value is 3.

Step 1: Simplify Inside the First Parentheses

First, evaluate the expression inside the first parentheses: \(6^{2} - 3^{3}\). Calculate each power separately: \(6^{2} = 36\) and \(3^{3} = 27\). Now subtract them: \(36 - 27 = 9\).

Step 2: Simplify Inside the Second Parentheses

Next, calculate the expression inside the second parentheses: \(4^{2} - 4\). First, calculate the power: \(4^{2} = 16\). Then subtract: \(16 - 4 = 12\).

Step 3: Perform the Multiplication

Multiply the result from Step 1 by 4: \(4 \times 9 = 36\).

Step 4: Perform the Division

Now divide the result from Step 3 by the result of Step 2: \(36 \div 12 = 3\).

Key Concepts

Exponents

Exponents are powerful mathematical tools that represent repeated multiplication. If you see an expression like \(a^b\), it means that the number \(a\) is multiplied by itself \(b\) times.
For instance, in the expression \(6^2\), the exponent 2 tells us to multiply 6 by itself: \(6 \times 6 = 36\). Similarly, for \(3^3\), the number 3 is multiplied by itself twice: \(3 \times 3 \times 3 = 27\).
Learning about exponents is essential because they simplify expressions and make calculations more manageable. Understanding how to handle them is crucial for solving complex math problems.

Parentheses

Parentheses in math act like a pair of umbrellas, grouping parts of an expression to indicate which operations should be performed first. This is crucial because it helps to clarify the order of operations, ensuring calculations are accurate.
In the original exercise, parentheses help us prioritize what to calculate first. For example, the expression \(4(6^2 - 3^3)\) tells us to solve \(6^2 - 3^3\) before multiplying by 4. This is due to the fundamental rule in mathematics that dictates operations within parentheses should always be performed first.
Using parentheses correctly can transform complex expressions into more straightforward, solvable problems, acting as a guide to navigate through intricate mathematical terrains.

Division

Division is the process of distributing a number into equal parts or groups. It is one of the basic arithmetic operations, alongside addition, subtraction, and multiplication.
In the context of the given problem, division is used at the end to conclude the sequence of operations. After simplifying the expressions inside the parentheses and multiplying, we arrive at a simplified result that needs to be divided. Specifically, the operation \(36 \div 12\) is performed here. This tells us we are dividing 36 into 12 equal parts, resulting in 3.
Understanding how to divide accurately is vital as it's often the step that brings together all prior calculations into a final, concise answer.

Subtraction

Subtraction is one of the most straightforward arithmetic operations, yet it plays an integral role in simplifying expressions. It involves taking one number away from another.
In the exercise example, subtraction is employed twice: once within the first set of parentheses, \(6^2 - 3^3\), and once within the second, \(4^2 - 4\). These operations are crucial steps in simplifying the components of the more extensive expression.
By mastering subtraction, we unravel complexities in problems by reducing numbers, ultimately getting closer to the needed simplified solution. Its importance in handling operations can't be overstated, as it assists in systematically working through problems to find answers effectively.