Problem 9 - Prealgebra and Introductory Algebra

Question

Simplify. $$2^{3}-(-3)^{2}+2$$

Step-by-Step Solution

Answer

The simplified version of the given expression is 1.

Step 1: Evaluate exponents

PEMDAS/BODMAS rule gets applied which stands for Parentheses first, Exponents (ie Powers and Square Roots, etc.) second, their acronym in order. Hence, apply 'E' from the order to evaluate the powers: \(2^{3} = 8\) and \((-3)^{2} = 9\). The expression becomes: \(8-9+2\)

Step 2: Perform subtraction

Continuing with PEDMAS/BODMAS rules to carry out the subtraction: \(8-9 = -1\). Substituting the value back into the expression, it now reads: \(-1+2\)

Step 3: Perform addition

Lastly, perform the addition operation: \(-1+2=1\)

Key Concepts

Simplify Expressions

Simplifying expressions is an essential skill in mathematics, as it involves reducing an expression to its most basic form. This process often requires the use of the order of operations to ensure accurate results. When simplifying expressions, it is crucial to tackle one operation at a time.

• Identify like terms that can be combined.
• Apply mathematical operations systematically.
• Use brackets and parentheses to clearly define parts of the expression.

In the context of our example, simplification involved breaking down the expression: \(2^{3}-(-3)^{2}+2\). By evaluating the powers first and tackling each element sequentially according to mathematical rules, we arrived at the simplified result.

PEMDAS

PEMDAS is an acronym representing the order of operations essential to simplifying expressions correctly. Each letter in PEMDAS stands for:

• P: Parentheses - Solve anything inside parentheses first.
• E: Exponents - Compute powers and roots.
• M and D: Multiplication and Division - Process these from left to right.
• A and S: Addition and Subtraction - Complete these from left to right.

By following these rules, we ensure that every mathematical expression is solved accurately. In our example,
we started with the exponents after confirming there were no parentheses impacting the expression \(2^{3}-(-3)^{2}+2\). Carefully following each step ensures clarity and precision in problem-solving.

Exponents

Exponents in mathematics represent repeated multiplication of a number by itself. For instance, \(2^{3}\) means multiplying 2 by itself 3 times, resulting in 8. Similarly, \((-3)^{2}\) is multiplying -3 by itself, yielding 9. Understanding exponents is crucial because they establish the power of numbers in expressions. Some key points about exponents include:

• When a negative number is raised to an even power, the result is positive.
• A negative number raised to an odd power remains negative.

Handling exponents correctly can dramatically change the result, as seen in our original expression. Properly simplifying exponents helps reduce complexity early in the problem.

Negative Numbers in Mathematics

Negative numbers can often be tricky, especially when involved in exponents or subtraction. Understanding how to handle them correctly is crucial for accurate mathematical operations. Some important tips include:

• When subtracting a negative number, it is equivalent to adding its positive counterpart.
• Pay attention to negative signs in expressions, especially during multiplication and powers.
• Negative powers invert the base number (although this was not directly relevant in our example).

In our original expression, we encountered a situation where \((-3)^{2}\) calculated the square of -3, resulting in 9, illustrating how negative signs can influence the outcome.