Solving Math Problems Value Of √a + 3∛b When A=9 And B=8

Hey there, math enthusiasts! Today, we're diving into an exciting mathematical expression that combines square roots and cube roots. We're going to break down the expression √a + 3∛b and find its value when a = 9 and b = 8. Buckle up, because we're about to embark on a journey of numbers and roots!

Unraveling the Expression: √a + 3∛b

First things first, let's get familiar with the expression itself. √a + 3∛b might look a bit intimidating at first glance, but don't worry, we'll tackle it step by step. The expression involves two main components: a square root (√a) and a cube root (∛b), with the cube root being multiplied by 3. To solve this, we need to understand what square roots and cube roots are all about.

Square Roots: A Quick Refresher

A square root of a number is a value that, when multiplied by itself, gives you the original number. For example, the square root of 9 (√9) is 3 because 3 * 3 = 9. Think of it as finding the side length of a square when you know its area. In our expression, we have √a, so we'll be looking for the square root of the value assigned to 'a'.

Cube Roots: Stepping into the Third Dimension

Now, let's talk about cube roots. A cube root of a number is a value that, when multiplied by itself three times, gives you the original number. For instance, the cube root of 8 (∛8) is 2 because 2 * 2 * 2 = 8. Imagine finding the side length of a cube when you know its volume. In our expression, we have ∛b, so we'll be finding the cube root of the value assigned to 'b'.

Understanding these concepts is crucial for tackling our expression. We're not just dealing with simple addition or multiplication; we're venturing into the world of roots, which adds a layer of complexity and excitement to the problem. But with a clear grasp of square roots and cube roots, we're well-equipped to solve this mathematical puzzle. So, let's move on to the next step: plugging in the values of 'a' and 'b' and see what happens!

Plugging in the Values: a = 9 and b = 8

Now comes the exciting part where we substitute the given values of a and b into our expression. We're told that a = 9 and b = 8. This means we're going to replace 'a' with 9 and 'b' with 8 in the expression √a + 3∛b. Let's do it!

Our expression now looks like this: √9 + 3∛8. See how we've simply replaced the variables with their corresponding numerical values? This is a fundamental step in solving algebraic expressions. It allows us to move from the abstract world of variables to the concrete realm of numbers. With these values in place, we can now focus on simplifying the expression and finding its final value. The substitution is like laying the groundwork for the solution, setting us up to perform the necessary calculations.

Calculating the Square Root of 9

Alright, let's tackle the first part of our expression: √9. We need to find the square root of 9, which means we're looking for a number that, when multiplied by itself, equals 9. Think about it – what number fits the bill? You guessed it – it's 3! Because 3 * 3 = 9, we can confidently say that √9 = 3. This is a crucial step in simplifying our expression. By finding the square root of 9, we've transformed a potentially complex term into a simple, manageable number. This is the beauty of mathematical operations – they allow us to break down problems into smaller, more digestible parts. So, with √9 resolved, let's move on to the next part of our expression: the cube root of 8.

Finding the Cube Root of 8

Next up, we have ∛8, which means we need to find the cube root of 8. Remember, the cube root is a number that, when multiplied by itself three times, equals 8. So, what number times itself three times gives us 8? If you thought of 2, you're spot on! Because 2 * 2 * 2 = 8, we know that ∛8 = 2. This is another significant step in simplifying our expression. Just like with the square root, finding the cube root allows us to replace a potentially tricky term with a simple number. This makes the overall expression much easier to work with. Now that we've found both the square root of 9 and the cube root of 8, we're in a great position to put everything together and find the final answer.

Putting It All Together: √9 + 3∛8

We've done the groundwork, we've found the square root of 9 (√9 = 3) and the cube root of 8 (∛8 = 2). Now it's time to bring it all together and solve the expression √9 + 3∛8. Remember, the order of operations matters! We need to handle the multiplication before the addition. So, let's start by multiplying 3 by the cube root of 8.

Multiplying 3 by the Cube Root of 8

We know that ∛8 = 2, so we need to calculate 3 * 2. This is a straightforward multiplication, and the answer is 6. So, 3∛8 = 6. Now our expression looks even simpler: 3 + 6. We've successfully reduced the complex expression to a simple addition problem. This is a testament to the power of breaking down problems into smaller steps. By tackling the square root, the cube root, and the multiplication individually, we've made the overall problem much more manageable. Now, let's finish it off with the final addition.

The Final Addition: 3 + 6

We're in the home stretch! We have the simplified expression 3 + 6. This is a basic addition problem, and the answer is, of course, 9. So, 3 + 6 = 9. Congratulations! We've successfully navigated through the expression and found the final value. By carefully following the order of operations and breaking down the problem into smaller, more manageable steps, we've arrived at the solution. This is a great example of how mathematical expressions can be solved with patience, attention to detail, and a solid understanding of the underlying concepts. Now that we have our final answer, let's take a look at the original question and see which answer choice matches our result.

The Grand Finale: Matching the Answer

We've crunched the numbers, we've simplified the expression, and we've arrived at our final answer: 9. Now, let's circle back to the original question and see which of the answer choices matches our result. The question presented us with four options:

A. 13 B. 12 C. 11 D. 9

Drumroll, please! The correct answer is D. 9. We did it! We successfully solved the mathematical expression and found the matching answer choice. This is a moment of triumph, a testament to our problem-solving skills and our understanding of square roots, cube roots, and the order of operations. By carefully breaking down the problem, tackling each component individually, and then putting it all together, we were able to arrive at the correct solution. So, give yourself a pat on the back – you've conquered this math challenge!

Key Takeaways: Mastering Mathematical Expressions

We've reached the end of our mathematical journey, and what a journey it has been! We've tackled square roots, cube roots, the order of operations, and finally, we've arrived at the correct answer. But beyond the specific problem we solved, there are some key takeaways that we can apply to any mathematical expression we encounter.

Breaking It Down: The Power of Simplification

One of the most important lessons we learned is the power of breaking down complex problems into smaller, more manageable parts. Instead of being intimidated by the entire expression √a + 3∛b, we tackled each component individually. We found the square root of 9, we found the cube root of 8, we performed the multiplication, and then we did the addition. By breaking the problem down, we made it much less daunting and much easier to solve.

Order Matters: Following the Rules

We also emphasized the importance of the order of operations. Remember, we had to multiply 3 by the cube root of 8 before we could add the square root of 9. Following the correct order is crucial for arriving at the correct answer. Ignoring the order of operations can lead to incorrect results and frustration. So, always keep the order of operations in mind when tackling mathematical expressions.

Practice Makes Perfect: Building Your Skills

Finally, we learned that practice makes perfect. The more we work with mathematical expressions, the more comfortable and confident we become. By solving problems like this one, we build our skills and our understanding of mathematical concepts. So, don't be afraid to tackle new challenges and keep practicing! With each problem you solve, you'll become a more skilled and confident mathematician.

So, there you have it! We've not only solved a mathematical expression but also learned some valuable lessons along the way. Remember to break down problems, follow the order of operations, and keep practicing. With these tools in your arsenal, you'll be well-equipped to conquer any mathematical challenge that comes your way!