Problem 90 - Elementary and Intermediate Algebra

Question

Evaluate each expression. $$ 3-\left[3^{3}+(3-1)^{3}\right] $$

Step-by-Step Solution

Answer

The evaluated expression is -32.

Step 1: Resolve the innermost bracket

Start by evaluating the expression inside the square brackets. Calculate \((3 - 1)^3\). Since \(3 - 1 = 2\), we have \((2)^3 = 8\).

Step 2: Resolve the expression inside the square bracket

Now replace \((3-1)^3\) with 8 in the expression: \(3^{3} + 8\). Calculate \(3^{3}\) which is \(27\). Therefore, the expression becomes \(27 + 8\).

Step 3: Simplify inside the bracket

Add the numbers inside the brackets: \(27 + 8 = 35\). Now, the entire expression simplifies to \(3 - [35]\).

Step 4: Complete the expression

Subtract 35 from 3: \(3 - 35 = -32\). This is the result of the original expression.

Key Concepts

Exponents

Exponents are an essential part of mathematics, especially in algebra and calculus. They indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(3^3\), the number 3 is the base and the exponent is 3. This means you have to multiply 3 by itself three times: \(3 \times 3 \times 3 = 27\).

Remember:

• An exponent of 2 is called 'squared' (\(n^2\)).
• An exponent of 3 is called 'cubed' (\(n^3\)).
• A base raised to the power of 1 is simply the base itself \(n^1 = n\).

Handling exponents can simplify complex arithmetic operations and make calculations more efficient. When dealing with expressions, evaluate any exponents first before proceeding to other operations. This is based on the order of operations rule, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Subtraction

Subtraction is one of the basic operations in mathematics and is the process of taking one quantity away from another. In the example expression \(3 - [35]\), subtraction is used to simplify the entire equation by removing 35 from 3.

Key points to remember about subtraction:

• It's the inverse operation of addition.
• The number being subtracted is called the subtrahend, and the number it is subtracted from is the minuend.
• When subtracting, if the minuend is smaller than the subtrahend, the result is a negative number.
• It is important to follow the order of operations to correctly carry out subtraction, especially in expressions with multiple operations.

By understanding these basic principles, you can handle more complex subtraction problems and avoid errors.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. They form the fundamental building blocks of mathematics. Understanding how to execute these operations, especially when combined in a single expression like the one given, is critical.

In the order of operations:

• Calculations inside parentheses or brackets go first.
• Exponents come next.
• Multiplication and division follow, from left to right.
• Addition and subtraction are last, also from left to right.

This specific order ensures that everyone evaluates mathematical expressions the same way. In the expression \(3 - \left[3^3 + (3 - 1)^3\right]\):

• First, solve inside the brackets (\([3^3 + (3-1)^3]\)).
• Calculate any exponential terms within.
• Convert the bracket expression to a single number.
• Finally, complete the subtraction from the initial number.

This systematic approach helps prevent confusion and mistakes while simplifying complex expressions effectively.