Question

For Exercises \(1-74,\) simplify. $$5[4+3(-2+1)]-(-4)^{2}$$

Short answer

Question: Simplify the expression: \(5[4 + 3(-2 + 1)] - (-4)^2\) Answer: -11

Step-by-step solution

Parentheses Inside the Brackets

First, we will simplify the expression inside the parentheses, which is \((-2 + 1)\). Since this is an addition operation, we can directly add the numbers, resulting in \(-2 + 1 = -1\). Now, the expression becomes: $$5[4 + 3(-1)] - (-4)^2$$

Simplify Operations Inside the Brackets

Now, we will simplify the operations inside the brackets. There is only one operation inside the brackets, which is \(3(-1)\). Multiplying \(3\) by \(-1\) results in \(-3\). The expression now becomes: $$5[4 - 3] - (-4)^2$$

Simplify the Brackets Further

Next, we will simplify the remaining operation inside the brackets, which is \(4 - 3\). Subtracting \(3\) from \(4\) results in \(1\). The expression now becomes: $$5[1] - (-4)^2$$

Evaluate the Exponent

Now that we have simplified the brackets, we move on to the next operation in the order of operations: the exponent. We need to evaluate \((-4)^2\). Squaring \(-4\) results in \((-4) * (-4) = 16\). The expression now becomes: $$5(1) - 16$$

Perform Multiplication

Next, we perform the multiplication operation: \(5(1)\). Multiplying \(5\) by \(1\) results in \(5\). The expression now becomes: $$5 - 16$$

Perform Subtraction

Finally, we perform the subtraction operation: \(5 - 16\). Subtracting \(16\) from \(5\) results in \(-11\). The simplified expression is: $$-11$$

Key concepts

Order of Operations

To accurately simplify algebraic expressions, we need to understand the order of operations, also known as PEMDAS. This acronym stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). It's crucial to follow this order to get the correct result.

For example, in the given exercise, parentheses must be addressed first before moving on to exponents. By following PEMDAS, we ensure that operations within parentheses, like \( -2 + 1 \), are completed before multiplying or dealing with exponents such as \( -4 \)^2. Failure to follow this sequence can lead to incorrect answers and misunderstandings in more complex problem-solving scenarios.

Evaluating Exponents

Evaluating exponents is another critical step in simplifying algebraic expressions. An exponent tells us how many times to multiply a number by itself. It is important to remember that \( -4 \)^2 is not the same as \( -4 * -4 \) which would be -16; it means that we multiply -4 by itself to get 16.

Using our example, \( -4 \)^2 must be evaluated after dealing with the parentheses. Recognizing that the exponent applies only to the -4 and not to the sign helps avoid mistakes. Always raise the number to the power indicated by the exponent before carrying out any multiplication or division with other terms.

Performing Multiplication and Division

After simplifying any parentheses and evaluating exponents, we move on to performing multiplication and division. These operations are of equal precedence and must be carried out from left to right as they appear in the expression. In our exercise, once we resolve the operations within brackets, we're left with multiplication, such as \( 5(1) \).

This step is fairly straightforward when dealing with single digits, but it's still important to do these operations in the correct sequence and apply them correctly when working with more complex expressions. Consistent practice in performing multiplication and division will prevent errors as algebraic expressions become more challenging.

Arithmetic Operations

The last step in simplifying algebraic expressions involves arithmetic operations: addition and subtraction. These should always be performed after parentheses, exponents, multiplication, and division are taken care of. As with multiplication and division, these operations are to be carried out from left to right.

In our problem, after multiplying 5 by 1 and evaluating the exponent \( -4 \)^2, we subtract 16 from 5. It can be easy to overlook this simple step, but completing the arithmetic operations correctly is essential to finalize your simplified expression. Remembering to combine like terms and follow the correct operation signs will lead to the correct simplified result of \( -11 \).