Evaluate A³ + B² When A = 2 And B = -1/3

Hey guys! Today, we're diving into a fun little math problem where we need to evaluate the expression a³ + b². This simply means we need to find the value of this expression when we know what a and b are. In this case, we're given that a equals 2 and b equals -1/3. Sounds like a plan? Let's get started!

Understanding the Expression

Before we jump into plugging in the numbers, let's quickly break down what the expression a³ + b² actually means. This is super important for avoiding any silly mistakes later on.

• a³ (a cubed): This means we need to multiply a by itself three times: a * a * a*. It's also known as "a to the power of 3".
• b² (b squared): This means we need to multiply b by itself two times: b * b*. It's also known as "b to the power of 2".
• The + sign: This simply means we need to add the result of a³ to the result of b². Easy peasy!

So, to recap, we're going to calculate a cubed, then calculate b squared, and finally add those two results together. Make sense? Great! Now, let's plug in those values.

Step-by-Step Evaluation

Okay, now for the exciting part! We know a = 2 and b = -1/3. Let's substitute these values into our expression a³ + b² and see what we get.

1. Calculate a³

First up, let's tackle a³. We know a is 2, so we need to calculate 2³. Remember, this means 2 * 2 * 2. Let's do it:

2 * 2 = 4

4 * 2 = 8

So, a³ which is 2³, equals 8. Awesome! We've got the first part down.

2. Calculate b²

Next, let's figure out b². We know b is -1/3, so we need to calculate (-1/3)². This means (-1/3) * (-1/3). Now, remember the rules for multiplying fractions and negative numbers:

• When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers).
• A negative number multiplied by a negative number gives a positive number.

So, let's do the math:

(-1/3) * (-1/3) = (1 * 1) / (3 * 3) = 1/9

Therefore, b², which is (-1/3)², equals 1/9. Fantastic! We've got the second part solved as well.

3. Add a³ and b²

We're almost there! Now we just need to add the results we got for a³ and b². We found that a³ = 8 and b² = 1/9. So, we need to calculate 8 + 1/9.

To add a whole number and a fraction, we need to express the whole number as a fraction with the same denominator as the other fraction. In this case, our denominator is 9. So, we need to express 8 as a fraction with a denominator of 9.

To do this, we multiply 8 by 9/9 (which is just 1, so we're not changing the value):

8 * (9/9) = 72/9

Now we can add the fractions:

72/9 + 1/9 = (72 + 1) / 9 = 73/9

So, 8 + 1/9 = 73/9.

4. Express as a Mixed Number (Optional)

We've got our answer as an improper fraction (the numerator is bigger than the denominator), which is perfectly fine! But sometimes, it's nice to express it as a mixed number (a whole number and a fraction).

To do this, we divide the numerator (73) by the denominator (9):

73 ÷ 9 = 8 with a remainder of 1

This means that 73/9 is equal to 8 whole numbers and 1/9. So, as a mixed number, 73/9 is 8 1/9.

Final Answer

Phew! We made it through all the steps. So, the final answer to the question of evaluating a³ + b² when a = 2 and b = -1/3 is 73/9 (or 8 1/9 as a mixed number).

Key Concepts and Takeaways

Let's recap the key concepts we used in this problem. This is super helpful for tackling similar problems in the future:

• Understanding exponents: Knowing what a³ and b² mean is fundamental. Remember, the exponent tells us how many times to multiply the base by itself.
• Substituting values: We replaced the variables (a and b) with their given numerical values. This is a crucial step in evaluating expressions.
• Order of operations: Although this problem was relatively simple, it's always good to remember the order of operations (PEMDAS/BODMAS). We dealt with the exponents first, and then the addition.
• Fractions: We worked with fractions, including squaring a fraction and adding a whole number to a fraction. Remember the rules for these operations!
• Improper fractions and mixed numbers: We learned how to express our answer as both an improper fraction and a mixed number.

Why is This Important?

You might be thinking, "Okay, that's cool, but why do we even need to know this stuff?" Well, evaluating expressions like this is a fundamental skill in algebra and mathematics in general. It's like the building block for more complex equations and problem-solving. You'll encounter this kind of thing all the time in higher-level math courses, in physics, engineering, and even in some areas of computer science.

Plus, it helps you develop your logical thinking and problem-solving skills, which are valuable in all aspects of life! So, even if you don't become a mathematician, this kind of practice is really beneficial.

Practice Problems

Want to test your understanding? Here are a few practice problems similar to the one we just solved:

1. Evaluate x² + y³ if x = 3 and y = -2.
2. Evaluate p³ - q² if p = -1 and q = 4.
3. Evaluate 2m² + n³ if m = 1/2 and n = 2.

Try solving these on your own. You can even try changing the numbers and making up your own problems! The more you practice, the more confident you'll become.

Conclusion

So, there you have it! We successfully evaluated the expression a³ + b² when a = 2 and b = -1/3. We broke down each step, discussed the key concepts, and even looked at why this is important. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So keep practicing, keep asking questions, and keep having fun with it!

I hope this explanation was helpful! Let me know if you have any other math questions. Happy calculating, guys!