Question

For Exercises \(1-74,\) simplify. $$-3 \sqrt{49}+|16 \div(-8)|-(28-40) \div(-3)$$

Short answer

Question: Simplify the given expression: $$-3\sqrt{49}+|-\frac{16}{8}|-\frac{28-40}{-3}$$. Answer: The simplified result of the given expression is $$-23$$.

Step-by-step solution

Square Root Calculation

First, we compute the square root of 49 and multiply it with -3. The square root of 49 is 7, so we have: $$-3 \cdot 7 = -21$$.

Division inside Absolute Value

Next, we perform the calculation inside the absolute value: $$\frac{16}{-8} = -2$$. Then, we need to find the absolute value of -2, which is: $$|-2| = 2$$.

Parentheses and Division

Now, we calculate the subtraction inside the parentheses: $$(28-40) = -12$$ Then, we divide this result by -3: $$\frac{-12}{-3} = 4$$.

Put Everything Together

Finally, we have all the simplified intermediate results, and it's time to put them all together. We get: $$-21 + 2 - 4$$.

Addition and Subtraction

We perform the addition first: $$(-21) + 2 = -19$$ And then, we perform the subtraction: $$-19 - 4 = -23$$. The simplified result of the given expression is $$-23$$.

Key concepts

Square Root Calculation

Understanding how to compute square roots is essential in simplifying algebraic expressions. Let's break down this concept. A square root asks the question: 'which number, when multiplied by itself, gives the original number?' For instance, when we see \( \sqrt{49} \), we are looking for a number that, multiplied by itself, equals 49.

In our exercise, the square root of 49 is simply 7 because \( 7 \times 7 = 49 \). However, it's important to remember that negative numbers don't have real square roots since a negative times a negative gives a positive result. After finding the square root, we then multiply it by -3, following the order of operations, which gives us -21. Practice identifying perfect squares and using a calculator when needed to help speed up the process of finding square roots.

Absolute Value

The concept of absolute value is best understood as the 'distance a number is from zero on the number line,' without considering direction. Therefore, the absolute value of a number is always non-negative. In our exercise, we are asked to find the absolute value of \( \frac{16}{-8} \), which is -2.

To visualize this, imagine the number -2 on the number line; it is 2 units away from zero. So, \( |-2| = 2 \). Whether the input is positive or negative, the output of the absolute value function will be positive. The absolute value bars act like a shield, converting any negative 'inside' into a positive before proceeding with further calculations. Remembering this property can help you simplify expressions involving absolute values with greater ease.

Order of Operations

When simplifying algebraic expressions, the order of operations is paramount. This is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the exercise, we follow this order to simplify the expression step by step. We start by addressing the operations within the parentheses (28-40) and perform the division that follows. After that, we follow up with the square root calculation and tackle the absolute value, and finally, we perform the addition and subtraction to arrive at the simplified result. By consistently applying the order of operations, you ensure that you get the correct result every time, avoiding common mistakes that can occur when operations are performed out of sequence.