Problem 50 - Elementary Algebra

Question

Simplify. $$ 4+3 *\left((-3)^{\wedge} 3+5^{\wedge} 2\right) / 6-2^{\wedge} 2 $$

Step-by-Step Solution

Answer

The simplified result of the expression is \(-1\).

Step 1: Evaluate Exponents

Calculate the exponents in the expression:\((-3)^3 = -27\)\(5^2 = 25\)\(2^2 = 4\)Substituting these values back into the expression, we have:\[4 + 3 \times \left( -27 + 25 \right) / 6 - 4\]

Step 2: Simplify Parentheses

Simplify inside the parentheses:\(-27 + 25 = -2\)Substitute back:\[4 + 3 \times (-2) / 6 - 4\]

Step 3: Perform Multiplication and Division

First, multiply and divide from left to right:\(3 \times (-2) = -6\)\(-6 / 6 = -1\)Substitute back:\[4 + (-1) - 4\]

Step 4: Perform Addition and Subtraction

Now, perform addition and subtraction from left to right:\(4 - 1 = 3\)\(3 - 4 = -1\)The simplified expression is:\(-1\)

Key Concepts

Exponents

Exponents are a concise way to express repeated multiplication of the same number. They appear in the format of a base number raised to a certain power. For example, when you see \((-3)^3\), it means multiplying \(-3\) three times by itself: \(-3\times -3 \times -3 = -27\). It is important to note the negative sign is included, hence the result is negative.

Exponents are a key part of algebraic simplification because they reduce complex multiplication operations into simpler terms that are easier to handle. Another example is \(5^2\), which equals \(5 \times 5 = 25\).

Below are some quick reminders when working with exponents:

• An exponent of 1 means the value is the base itself, as \(x^1 = x\).
• An exponent of 0 always results in the value 1, as \(x^0 = 1\), except when \(x = 0\).

Understanding and correctly calculating exponents is foundational when simplifying algebraic expressions.

Order of Operations

In algebra, the order of operations is a set of rules that determine the correct sequence for solving mathematical expressions. Memorizing and using these rules is crucial for correctly simplifying expressions, ensuring every problem is solved consistently. The standard order is often remembered by the acronym PEMDAS:

• **P** Parentheses - Solve expressions inside parentheses first.
• **E** Exponents - Calculate all exponential terms next.
• **MD** Multiplication and Division - Work from left to right.
• **AS** Addition and Subtraction - Finally, perform these operations, also from left to right.

In our problem, after calculating the exponents, we simplify the expression inside the parentheses: \(-27 + 25 = -2\).

Following this, multiplication \(3 \times -2\) and division \(-6/6\) come next, followed by addition and subtraction. Always follow the order strictly to get accurate results.

Evaluating Expressions

Evaluating expressions involves simplifying them step by step until you reach a single value. It's an essential skill in algebra, involving the use of several rules and mathematical concepts.

When evaluating the given expression: \[4 + 3 \times \left((-3)^3 + 5^2\right) / 6 - 2^2\] We first deal with the exponents, as explained earlier. Next, we handle the operations within the parentheses (\(-27 + 25\)), simplify by multiplying and dividing, and finally execute any addition or subtraction steps.

When evaluating, always stick to the sequence dictated by the order of operations. Each step must be completed in the correct sequence before moving to the next. Breaking down the process like this makes even complex-looking expressions more manageable and less intimidating.