Problem 12 - Fundamentals of Mathematics

Question

Determine the value of each of the following. \(\\{6-[24 \div(4 \cdot 2)]\\}^{3}\)

Step-by-Step Solution

Answer

The value is 27.

Step 1: Solve the Expression Inside the Parentheses

Identify and solve the operation inside the innermost parentheses first. In the expression \(4 \cdot 2\), multiply 4 by 2 to get 8. Substitute back into the expression: \[6 - [24 \div 8]^{3}\]

Step 2: Division Inside the Brackets

Proceed with the division inside the brackets. Divide 24 by 8 to get 3. Substitute back into the expression:\[6 - 3^{3}\]

Step 3: Complete the Subtraction

Subtract 3 from 6. This results in:\[3^{3}\]

Step 4: Calculate the Power

Finally, calculate the power of 3 to the 3rd power. Multiply 3 by itself twice: \(3 \times 3 = 9\) and then \(9 \times 3 = 27\).

Step 5: Final Result

Thus, the value of the expression is 27.

Key Concepts

Exponents

Exponents are a mathematical operation, representing repeated multiplication of the same number. In an expression like \(3^3\), the base number is 3, and the exponent is also 3. This means you multiply 3 by itself three times.

• Step 1: Calculate \(3 \times 3 = 9\).
• Step 2: Multiply the result by 3 again: \(9 \times 3 = 27\).

Using exponents helps condense repeated multiplication into a more compact form. This is especially useful in various mathematical computations and can simplify expressions significantly.

Parentheses

Parentheses are used in mathematical expressions to group numbers and operations together, indicating which operations should be performed first. When solving an expression, always start with calculations inside the parentheses.
For example, in the expression \(6 - [24 \div (4 \cdot 2)]^3\), focus first on the innermost parentheses: \(4 \cdot 2\).

• Calculate \(4 \cdot 2\) to obtain 8.

This simplifies the expression inside the brackets, which directs what operations you should do next. Remember, handling what's inside parentheses first is a key part of the order of operations.

Multiplication and Division

Both multiplication and division are fundamental arithmetic operations in math. They are performed left to right, based on their order of appearance in an expression, after operations inside parentheses.
In the expression \(6 - [24 \div 8]^3\), after simplifying the content within parentheses:

• Perform the division: Divide 24 by 8 to get 3.

Always follow the correct order to maintain the balance of the expression. Mistakes in this step can lead to incorrect results, so ensure you work systematically.

Subtraction

Subtraction is the arithmetic operation of finding the difference between numbers. After simplifying terms within parentheses and performing any multiplication or division, subtraction often follows.
In our example, once you've dealt with the multiplication and division, the expression simplifies to \(6 - 3^3\).

• Subtract 3 from 6, which results in 3.

This step may seem simple, but focusing on precision ensures the final answer is accurate, especially when an exponent follows the subtraction step.