Evaluate √(x⁴) - Y² When X = 3 And Y = -6

Hey everyone! Let's dive into a fun math problem today. We're going to evaluate the expression √(x⁴) - y² given the values x = 3 and y = -6. It might look 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 numbers, let's take a moment to understand the expression √(x⁴) - y² itself. This is crucial for avoiding common mistakes and ensuring we get the correct answer. The expression consists of two main parts: √(x⁴) and y², which are then subtracted from each other. The first part involves a square root and an exponent, while the second part is simply a variable squared.

Breaking Down √(x⁴)

The term √(x⁴) might seem a bit complex, but it's actually quite manageable. Remember that the 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 is 3 because 3 * 3 = 9. Now, when we have x⁴, this means x raised to the power of 4, which is x multiplied by itself four times (x * x * x * x*). So, we're essentially looking for the square root of x multiplied by itself four times. A key property to remember here is that √(a²) = |a|, the absolute value of a. This is because squaring a number always results in a non-negative value, and the square root function returns the non-negative root. Therefore, √(x⁴) can be simplified to |x²|, which is simply x² since x² is always non-negative. This simplification is a game-changer because it makes the expression much easier to work with. We've effectively transformed a seemingly complicated square root and exponent combination into a simple square. This kind of simplification is a common strategy in algebra, and mastering it can make solving problems much smoother.

Understanding y²

The second part of our expression is y², which is much more straightforward. It simply means y multiplied by itself (y * y). For example, if y were 2, then y² would be 2 * 2 = 4. If y were -2, then y² would be -2 * -2 = 4. Notice that squaring a negative number results in a positive number. This is because a negative times a negative is always a positive. This basic understanding of squaring numbers is essential for correctly evaluating the expression, especially when dealing with negative values like y = -6. The simplicity of y² doesn't mean it's any less important than the first part of the expression. It plays a crucial role in the final result, and we need to handle it with the same care and precision as we did with √(x⁴).

Substituting the Values

Now that we've dissected the expression and understand each part individually, it's time to plug in the given values for x and y. This is where the fun really begins! We're given that x = 3 and y = -6. Our goal is to substitute these values into the simplified expression and then perform the necessary calculations to find the final answer. This process is a fundamental skill in algebra, and it's something you'll use again and again in more advanced math problems. Let's take it one step at a time to ensure we don't make any silly mistakes.

Substituting x = 3 into √(x⁴)

We've already simplified √(x⁴) to x². So, when we substitute x = 3, we get 3². This means 3 multiplied by itself, which is 3 * 3 = 9. So, the value of √(x⁴) when x = 3 is 9. It's crucial to remember the order of operations (PEMDAS/BODMAS), which dictates that we handle exponents before other operations like subtraction. This is why we squared the value of x before moving on to the next part of the expression. A common mistake is to try and take the square root before applying the exponent, which would lead to an incorrect result. By simplifying the expression first and then substituting, we've made the calculation much cleaner and less prone to errors. This approach is a good habit to develop for any algebraic problem.

Substituting y = -6 into y²

Next, we substitute y = -6 into y². This means we need to calculate (-6)². Remember that squaring a negative number results in a positive number. So, (-6)² is -6 * -6 = 36. This is a critical point to remember, as forgetting the negative sign would lead to a completely different answer. The value of y² when y = -6 is 36. The fact that squaring a negative number results in a positive number is a fundamental concept in mathematics. It's essential for understanding various topics, including quadratic equations, complex numbers, and even some areas of calculus. By correctly applying this rule here, we're ensuring that our final answer will be accurate. Mistakes with negative signs are very common, so double-checking these calculations is always a good idea.

Evaluating the Final Expression

Now that we've calculated the values of √(x⁴) and y² separately, we can put them back into the original expression and find the final answer. We found that √(x⁴) = 9 when x = 3, and y² = 36 when y = -6. So, our expression becomes 9 - 36. This is a simple subtraction problem, but it's important to get the signs right. We're subtracting a larger number (36) from a smaller number (9), so the result will be negative. 9 - 36 = -27. Therefore, the value of the expression √(x⁴) - y² when x = 3 and y = -6 is -27.

Double-Checking the Answer

It's always a good idea to double-check your answer, especially in math problems. One way to do this is to go back through each step and make sure you haven't made any mistakes. Did we correctly simplify √(x⁴)? Yes, we simplified it to x². Did we correctly substitute the values for x and y? Yes, we substituted x = 3 and y = -6. Did we correctly calculate 3² and (-6)²? Yes, we found 3² = 9 and (-6)² = 36. Did we correctly perform the subtraction 9 - 36? Yes, we got -27. Another way to double-check is to use a calculator, especially for more complex calculations. While this problem wasn't overly complex, using a calculator to verify our results can give us extra confidence. By double-checking our work, we're ensuring that we're submitting the most accurate answer possible. This habit of verifying solutions is crucial in mathematics and many other fields.

Conclusion

So, guys, we've successfully evaluated the expression √(x⁴) - y² when x = 3 and y = -6. We broke down the expression, simplified it, substituted the values, and performed the calculations. The final answer is -27. Remember, the key to solving these kinds of problems is to take them step by step, understand each part of the expression, and double-check your work. Math can be fun and challenging, and I hope this explanation helped you understand this problem a little better. Keep practicing, and you'll become a math whiz in no time!