Evaluate Expression: √[a³-7] + |b| For A=2, B=-4

Nov 10, 2025 by Admin 49 views

Evaluating the Expression √[a³-7] + |b| When a=2 and b=-4

Hey guys! Today, we're diving into a cool math problem where we need to evaluate an expression. Specifically, we're going to figure out the value of ${\sqrt{a^3-7}+|b|}$ when ${a=2}$ and ${b=-4}$ . This might sound a bit intimidating at first, but trust me, we'll break it down step by step and make it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Expression

Before we jump into plugging in the values, let's take a closer look at the expression itself: ${\sqrt{a^3-7}+|b|}$ . This expression has two main parts: a square root part and an absolute value part.

First, we have ${\sqrt{a^3-7}}$ . This means we need to:

Cube a: That's ${a}$ multiplied by itself three times ( ${a * a * a}$ ).
Subtract 7: We take the result from step 1 and subtract 7 from it.
Take the square root: Finally, we find the square root of the value we got in step 2.

Next, we have ${|b|}$ . This represents the absolute value of ${b}$ . The absolute value of a number is its distance from 0, so it's always non-negative. For example, the absolute value of -4 is 4, and the absolute value of 4 is also 4.

Our main keyword here is evaluating expressions, which is a fundamental concept in algebra. It involves substituting given values for variables and then simplifying the expression using the order of operations. So, let's keep this keyword in mind as we move forward.

Breaking Down the Components

Let's dive a bit deeper into each component of the expression. Understanding each part individually will make the overall evaluation much smoother. The first part, ${\sqrt{a^3-7}}$ , involves cubing, subtraction, and then finding the square root. These are all essential arithmetic operations that we use frequently in mathematics.

Cubing: The term ${a^3}$ means ${a}$ raised to the power of 3, which is ${a * a * a}$ . When we substitute ${a = 2}$ , we get ${2^3 = 2 * 2 * 2 = 8}$ .
Subtraction: After cubing ${a}$ , we subtract 7 from the result. So, we have ${8 - 7 = 1}$ .
Square Root: The final step for this part is to find the square root of the result. The square root of 1 is 1, because ${1 * 1 = 1}$ . Therefore, ${\sqrt{1} = 1}$ .

The second part of the expression is ${|b|}$ , which represents the absolute value of ${b}$ . Understanding absolute value is crucial here. The absolute value of a number is its distance from 0 on the number line. It's always a non-negative value. For example:

The absolute value of 5 (written as ${|5|}$ ) is 5.
The absolute value of -5 (written as ${|-5|}$ ) is also 5.

In our case, ${b = -4}$ , so the absolute value of ${b}$ is ${|-4| = 4}$ . This is because -4 is 4 units away from 0 on the number line.

Order of Operations

When evaluating expressions, it's crucial to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the operations in the correct sequence. In our expression, we have exponents (cubing), subtraction, square root, and absolute value. Although there aren't any parentheses in our specific expression, it's always a good practice to keep PEMDAS in mind for more complex problems.

By understanding each component and the order of operations, we set ourselves up for a successful evaluation. Now that we've dissected the expression, let's move on to the next step: substituting the given values.

Substituting the Values

Now that we understand the expression ${\sqrt{a^3-7}+|b|}$ , let's plug in the given values for ${a}$ and ${b}$ . We're told that ${a=2}$ and ${b=-4}$ . Substituting these values into the expression, we get:

${\sqrt{(2)^3-7}+|-4|}$

This looks much more manageable now, right? We've replaced the variables with their numerical values, and now we just need to simplify using the order of operations. Remember, our main keyword is still evaluating expressions, and substitution is a key step in this process.

Step-by-Step Substitution

Let's go through the substitution process step by step to make sure everything is crystal clear. This meticulous approach is particularly helpful when dealing with more complex expressions. When evaluating expressions, accuracy is paramount, and breaking the process into smaller steps reduces the chance of errors.

Replace a with 2: In the expression ${\sqrt{a^3-7}}$ , we replace ${a}$ with 2, which gives us ${\sqrt{(2)^3-7}}$ .
Replace b with -4: In the term ${|b|}$ , we replace ${b}$ with -4, resulting in ${|-4|}$ .

By performing these substitutions, we transform the original expression with variables into a numerical expression that we can simplify. The substituted expression, ${\sqrt{(2)^3-7}+|-4|}$ , is now ready for the next phase: simplification.

Importance of Accurate Substitution

In the realm of evaluating expressions, accurate substitution is the cornerstone of obtaining the correct result. A minor error in substitution can cascade through the entire process, leading to a drastically different answer. Therefore, it is essential to double-check every substitution to ensure precision.

Imagine, for instance, if we mistakenly substituted ${a = 3}$ instead of ${a = 2}$ . The expression ${\sqrt{a^3-7}}$ would become ${\sqrt{(3)^3-7} = \sqrt{27-7} = \sqrt{20}}$ , which is a completely different value from what we would get with the correct substitution. This highlights the critical role that accurate substitution plays in evaluating expressions successfully.

Now that we've successfully substituted the values, the next step is to simplify the expression. We will begin by simplifying the terms inside the square root and the absolute value, adhering to the order of operations.

Simplifying the Expression

Alright, we've substituted the values, and now it's time to simplify! We have the expression ${\sqrt{(2)^3-7}+|-4|}$ . Remember PEMDAS? We'll use that as our guide to make sure we simplify in the right order. Our main keyword here is still evaluating expressions, and simplification is a crucial part of that.

Simplifying Inside the Square Root

Let's start with the square root part: ${\sqrt{(2)^3-7}}$ . According to PEMDAS, we need to deal with the exponent first:

(2)³: This means 2 * 2 * 2, which equals 8. So, we now have ${\sqrt{8-7}}$ .

Next, we perform the subtraction:

8 - 7: This equals 1. So, our expression inside the square root simplifies to ${\sqrt{1}}$ .

Simplifying the Absolute Value

Now let's tackle the absolute value part: ${|-4|}$ . Remember, the absolute value of a number is its distance from 0, so it's always non-negative.

|-4|: The absolute value of -4 is 4.

Putting It All Together

Now we've simplified both parts of the expression. We have:

${\sqrt{1} + 4}$

We're almost there! We just need to evaluate the square root and then add the numbers together.

Step-by-Step Simplification Breakdown

For a clearer understanding, let's break down the simplification process step by step. When evaluating expressions, a methodical approach can significantly reduce the risk of making errors. Each step is a mini-operation that contributes to the overall simplification process.

Evaluate the Exponent: In the term ${\sqrt{(2)^3-7}}$ , the first operation is to evaluate the exponent. ${2^3}$ means 2 raised to the power of 3, which is ${2 * 2 * 2 = 8}$ . The expression becomes ${\sqrt{8-7}}$ .
Perform Subtraction: Inside the square root, we subtract 7 from 8: ${8 - 7 = 1}$ . The expression simplifies to ${\sqrt{1}}$ .
Evaluate Absolute Value: The absolute value of -4, denoted as ${|-4|}$ , is the distance of -4 from 0, which is 4. So, ${|-4| = 4}$ .
Combine Simplified Terms: Now, we combine the simplified terms: ${\sqrt{1} + 4}$ .

This detailed breakdown illustrates how each operation is performed in sequence, following the order of operations. By simplifying each component separately, we make the overall expression easier to manage. Now, we are ready for the final evaluation.

Final Evaluation

We've simplified the expression to ${\sqrt{1} + 4}$ . Now it's the final stretch! We just need to evaluate the square root and then add. Our main keyword is evaluating expressions, and this is the culmination of all our hard work!

Evaluating the Square Root

The square root of 1 is simply 1 because 1 * 1 = 1. So, ${\sqrt{1} = 1}$ .

Adding the Values

Now we add the result to the remaining value:

1 + 4: This equals 5.

The Final Answer

So, the value of the expression ${\sqrt{a^3-7}+|b|}$ when ${a=2}$ and ${b=-4}$ is 5. Awesome! We did it!

Importance of the Final Step

The final step in evaluating expressions is crucial because it brings together all the intermediate results to arrive at the final answer. This step often involves a simple arithmetic operation, such as addition or subtraction, but it is just as important as the earlier, more complex operations. Accuracy in this final step is essential to ensure that the entire evaluation process yields the correct result.

For instance, in our example, we had simplified the expression to ${\sqrt{1} + 4}$ . If we correctly evaluate ${\sqrt{1}}$ as 1 but then mistakenly add 3 instead of 4, we would arrive at an incorrect final answer. This highlights the significance of paying close attention to detail even in the final stages of evaluating expressions.

By meticulously evaluating each component and combining them accurately, we can confidently arrive at the final result. This final evaluation solidifies our understanding of the expression and confirms our proficiency in evaluating expressions.

Conclusion

Guys, we've successfully evaluated the expression ${\sqrt{a^3-7}+|b|}$ when ${a=2}$ and ${b=-4}$ . We broke down the problem step by step, from understanding the expression to substituting values, simplifying, and finally, evaluating. Remember, the main keyword we focused on was evaluating expressions, and we've seen how important each step is in this process.

Key Takeaways

Here are some key takeaways from our journey:

Understanding the Expression: Knowing what each part of the expression means is crucial.
Substituting Values: Replacing variables with their given values accurately is essential.
Simplifying Using PEMDAS: Following the order of operations ensures we simplify correctly.
Final Evaluation: Adding or subtracting the final values to get the answer.

Practice Makes Perfect

Evaluating expressions is a skill that gets better with practice. The more you work on these types of problems, the more comfortable and confident you'll become. Try changing the values of ${a}$ and ${b}$ and see what you get!

So, there you have it! We've tackled this math problem together, and hopefully, you feel a bit more confident about evaluating expressions. Keep practicing, and you'll be a math whiz in no time! Thanks for joining me, and see you in the next one! This exercise demonstrated our ability to apply arithmetic operations in the correct order and reinforced the concept of absolute value. By solving such problems, we improve our mathematical aptitude and problem-solving skills.