Square Root Of 8 + √60: Step-by-Step Solution
Hey everyone! Today, we're diving into an exciting mathematical journey: figuring out the square root of 8 + √60. Sounds a bit intimidating, right? But trust me, we'll break it down step by step, and you'll see it's totally manageable. Think of it as a puzzle – a fun one! So, grab your thinking caps, and let's get started!
Why This Matters
You might be wondering, “Why should I care about the square root of 8 + √60?” Well, math isn't just about numbers; it's about problem-solving. These kinds of problems pop up in various fields, from engineering and physics to computer science and even finance. Mastering this skill sharpens your analytical thinking, making you a better problem-solver in general. Plus, it's kinda cool to impress your friends with your math skills, right?
Breaking Down the Problem
1. Understanding the Basics of Square Roots
First, let’s ensure we are on the same page regarding square roots. 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 is 3 because 3 * 3 = 9. When we see something like √60, it means we're looking for a number that, when squared, equals 60. This isn't a whole number, but don't worry, we'll deal with it.
The key to tackling complex square roots like √(8 + √60) is to simplify them into a form we can easily understand and work with. The expression inside the main square root, 8 + √60, is a combination of a whole number and another square root. Our goal is to rewrite this in a way that allows us to "unwrap" the outer square root. Think of it like peeling an onion – we need to get to the core by removing the layers one by one.
2. The Nested Square Root Challenge
What makes √(8 + √60) a bit tricky is that we have a square root inside another square root – a nested square root. These nested structures often look intimidating, but there's a neat trick to handle them. We aim to express the entire expression inside the outer square root as a perfect square. Why? Because the square root of a perfect square is a whole number, which simplifies things dramatically. For instance, √(9) is 3 because 9 is a perfect square (3 * 3). Similarly, √((a + b)²) simplifies to a + b.
Our strategy involves recognizing that 8 + √60 looks like it might be the result of squaring a binomial expression involving square roots. In simpler terms, we're guessing that it might be in the form of (√x + √y)², where x and y are numbers we need to find. If we can find these numbers, we can rewrite the original problem in a much simpler form.
3. Setting Up the Equation
So, let's assume that √(8 + √60) can be written in the form √x + √y. Squaring both sides of this assumption will help us reveal the values of x and y. When we square (√x + √y), we use the formula (a + b)² = a² + 2ab + b². Applying this, we get:
(√x + √y)² = (√x)² + 2(√x)(√y) + (√y)² = x + 2√(xy) + y
Now, we want this to equal 8 + √60. This gives us a crucial equation to work with. We're trying to match the form x + 2√(xy) + y with 8 + √60. This means we need to find values for x and y that satisfy this relationship. The next step involves carefully comparing the terms and setting up a system of equations to solve.
4. Matching the Terms
Now, let's dive into matching the terms from our squared expression with the original expression. We have: x + 2√(xy) + y = 8 + √60 We can split this into two parts by equating the non-square root parts and the square root parts separately. This is a common strategy in algebra – breaking down one complex equation into simpler, manageable pieces.
First, let's look at the parts without the square roots. On the left side, we have x + y, and on the right side, we have 8. So, our first equation is: x + y = 8 This tells us that the sum of the two numbers, x and y, must be 8. Now, let's tackle the square root parts. On the left side, we have 2√(xy), and on the right side, we have √60. To equate these, we can write: 2√(xy) = √60 This equation involves the square roots, which means we'll need to do some further manipulation to solve for x and y. But don't worry, we're making good progress! We've transformed our initial problem into two simpler equations, and that's a big step forward.
5. Simplifying the Square Root Equation
Let’s simplify the square root equation we derived in the previous step: 2√(xy) = √60 To make things easier, our goal here is to isolate the √(xy) term. We can do this by dividing both sides of the equation by 2: √(xy) = √60 / 2 Now, we have a square root on both sides, but the right side looks a bit messy with the division. To clean this up, we can bring the 2 inside the square root. Remember that √a / b is not the same as √(a / b), but we can rewrite √60 / 2 in a useful way. Think of 2 as √(4), since √(4) = 2. So, we can rewrite the equation as: √(xy) = √60 / √(4) Now that we have square roots in the numerator and the denominator, we can combine them under a single square root: √(xy) = √(60 / 4) This simplifies things quite a bit. We just need to perform the division inside the square root.
6. Solving for xy
Continuing from where we left off, we had: √(xy) = √(60 / 4) Let's perform the division inside the square root: √(xy) = √(15) Now, to get rid of the square root on both sides, we can square both sides of the equation. This is a common technique when dealing with square roots because squaring a square root cancels it out. So, we square both sides: (√(xy))² = (√15)² This simplifies to: xy = 15 Great! We now have the product of x and y. This, combined with our earlier equation (x + y = 8), gives us a system of equations that we can solve for x and y. We're getting closer to finding our values, guys!
7. Forming a System of Equations
Alright, let’s take a step back and look at what we’ve accomplished. We’ve transformed our initial complex square root problem into a manageable system of equations. We have two equations:
- x + y = 8 (The sum of x and y is 8)
- xy = 15 (The product of x and y is 15)
This is a classic setup for solving for two variables. There are a couple of ways we can tackle this. One common method is substitution, where we solve one equation for one variable and then substitute that expression into the other equation. Another method involves recognizing a pattern or relationship that can help us find the values more directly. In this case, we can think about numbers that add up to 8 and multiply to 15. Often, with relatively simple numbers like these, you might be able to guess the solution. But let’s go through a more systematic approach using substitution so you can see the general method.
8. Solving by Substitution
Let's use the substitution method to solve our system of equations. We have:
- x + y = 8
- xy = 15
First, we'll solve the first equation for one of the variables. Let's solve for y: y = 8 - x Now, we'll substitute this expression for y into the second equation: x(8 - x) = 15 This gives us a single equation in terms of x, which we can solve. Let's expand and rearrange the equation: 8x - x² = 15 Move everything to one side to set the equation to zero: x² - 8x + 15 = 0 Now we have a quadratic equation. Time to solve it! There are several ways to solve a quadratic equation, such as factoring, using the quadratic formula, or completing the square. Factoring is often the quickest method if the quadratic expression can be factored easily.
9. Solving the Quadratic Equation
We've arrived at the quadratic equation: x² - 8x + 15 = 0 To solve this, we'll try factoring it. Factoring involves finding two numbers that multiply to give the constant term (15) and add up to give the coefficient of the linear term (-8). In this case, we're looking for two numbers that multiply to 15 and add up to -8. Those numbers are -3 and -5 because (-3) * (-5) = 15 and (-3) + (-5) = -8. So, we can factor the quadratic equation as: (x - 3)(x - 5) = 0 Now, for the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x: x - 3 = 0 or x - 5 = 0 Solving these gives us two possible values for x: x = 3 or x = 5 We have two potential solutions for x. Now we need to find the corresponding values for y. Remember, we have the equation y = 8 - x.
10. Finding the Values of y
We've found two possible values for x from our quadratic equation: x = 3 or x = 5 Now we need to find the corresponding values for y. We have the equation: y = 8 - x Let’s substitute each value of x into this equation.
- If x = 3: y = 8 - 3 = 5
- If x = 5: y = 8 - 5 = 3
So, we have two pairs of solutions: (x = 3, y = 5) and (x = 5, y = 3). Notice that the values are just swapped. This makes sense given our original equations, where x and y are interchangeable in the system.
Now that we have the values of x and y, we can go back to our original assumption that √(8 + √60) can be written in the form √x + √y. This is where we tie everything together and finally solve the problem!
Putting It All Together
11. Substituting Back into the Original Assumption
Okay, we're in the home stretch! We've found the values of x and y, which were the missing pieces of our puzzle. Remember, we assumed that: √(8 + √60) = √x + √y We found two pairs of solutions for x and y: (3, 5) and (5, 3). Let’s use the first pair, x = 3 and y = 5 (it doesn’t matter which pair we use since addition is commutative, meaning the order doesn't change the result).
Substitute these values into our assumption: √(8 + √60) = √3 + √5 This is our solution! We've successfully expressed the square root of 8 + √60 in a simpler form. It looks much cleaner and easier to understand, right? This shows the power of breaking down complex problems into smaller, more manageable steps.
12. Verifying the Solution
It's always a good idea to verify our solution to make sure we haven't made any mistakes along the way. We found that: √(8 + √60) = √3 + √5 To verify, we can square both sides of the equation and see if they are equal. Squaring the right side (√3 + √5) means multiplying it by itself: (√3 + √5)² = (√3 + √5)(√3 + √5) We can use the FOIL method (First, Outer, Inner, Last) or the formula (a + b)² = a² + 2ab + b² to expand this. Let’s use the formula: (√3 + √5)² = (√3)² + 2(√3)(√5) + (√5)² Now, simplify each term: (√3)² = 3 2(√3)(√5) = 2√15 (√5)² = 5 So, we have: (√3 + √5)² = 3 + 2√15 + 5 Combine the whole numbers: (√3 + √5)² = 8 + 2√15 Now, remember that √60 can be simplified. Since 60 = 4 * 15, we have √60 = √(4 * 15) = √4 * √15 = 2√15. So, our original expression inside the square root was 8 + √60, which is the same as 8 + 2√15. Our result matches the original expression! This confirms that our solution is correct. We did it, guys!
Conclusion: The Beauty of Mathematical Problem-Solving
So, there you have it! We've successfully navigated the tricky waters of nested square roots and found that √(8 + √60) = √3 + √5. This wasn't just about crunching numbers; it was about understanding the underlying principles, breaking down the problem, and applying the right techniques. We used algebra, simplification, and a bit of clever thinking to reach our solution.
The journey through this problem highlights the beauty of mathematical problem-solving. Each step built upon the previous one, and we transformed a seemingly complex problem into a series of manageable tasks. This is a skill that translates far beyond mathematics. The ability to break down problems, identify key components, and apply logical steps is invaluable in almost any field or endeavor.
Remember, guys, math isn't just about memorizing formulas; it's about developing a way of thinking. By tackling problems like this, you're sharpening your analytical skills and building a powerful toolset for tackling challenges in all areas of life. Keep practicing, keep exploring, and most importantly, keep having fun with math!
