Solving A Radical Expression: Step-by-Step Guide

Nov 2, 2025 by Admin 49 views

Hey guys! Let's break down this radical expression step by step. We're going to simplify each part and then put it all together. Don't worry; it's not as scary as it looks!

Breaking Down the Expression

Our mission, should we choose to accept it (and we do!), is to evaluate: √(625/81) : √(25/9) + 3(1/3) : √(1.(7)) - √36 : √(6^4)*. To make this easier, we'll tackle each radical and fraction individually.

1. Understanding the Expression

Before we dive into the nitty-gritty calculations, let's get a bird's-eye view of the expression we're dealing with. We have a series of operations involving square roots, fractions, division, addition, and subtraction. The key here is to follow the order of operations (PEMDAS/BODMAS) and simplify each term carefully. This means we'll first deal with the square roots, then handle the division and multiplication, and finally take care of the addition and subtraction. By breaking the expression down into smaller, manageable chunks, we can avoid making mistakes and keep the process organized. Remember, patience and attention to detail are our best friends in this mathematical journey!

2. Calculate √(625/81)

Let's start with the first term: √(625/81). We need to find the square root of both 625 and 81 separately. The square root of 625 is 25 because 25 * 25 = 625. The square root of 81 is 9 because 9 * 9 = 81. Therefore, √(625/81) = 25/9. This is our first simplified term, and we're off to a great start! Keep in mind that understanding perfect squares is crucial here. If you're not familiar with them, it might be helpful to brush up on your multiplication tables or use a calculator to quickly find the square roots. The more comfortable you are with recognizing perfect squares, the faster and more accurately you'll be able to simplify these types of expressions.

3. Calculate √(25/9)

Next up, we have √(25/9). This one is pretty straightforward. The square root of 25 is 5 (since 5 * 5 = 25), and the square root of 9 is 3 (since 3 * 3 = 9). So, √(25/9) = 5/3. See? We're making progress! Recognizing these perfect squares makes the calculation much smoother. If you encounter numbers that aren't perfect squares, you might need to estimate or use a calculator, but in this case, we're lucky to have nice, clean square roots. This step reinforces the importance of knowing your basic square roots, as it can save you a lot of time and effort when simplifying expressions.

4. Calculate 3*(1/3)

Now, let's tackle 3(1/3). This is simply 3 multiplied by 1/3. When you multiply a whole number by a fraction, you can think of it as multiplying the whole number by the numerator (the top part of the fraction) and then dividing by the denominator (the bottom part of the fraction). So, 3 * 1 = 3, and then we divide by 3, which gives us 3/3 = 1. Therefore, 3(1/3) = 1. This is a basic arithmetic operation, but it's essential to get it right. Remember, any number multiplied by its reciprocal (in this case, 3 and 1/3) will always equal 1. This principle can be helpful in simplifying more complex expressions as well.

5. Calculate √(1.(7))

Here we have √(1.(7)) which means the 7 repeats infinitely. This can be written as 1 and 7/9, which is 16/9. Taking the square root of 16/9 gives us √(16/9). The square root of 16 is 4 (since 4 * 4 = 16), and the square root of 9 is 3 (since 3 * 3 = 9). Thus, √(16/9) = 4/3. Converting repeating decimals to fractions is a handy trick to remember. In this case, recognizing that 1.(7) is equivalent to 16/9 allows us to easily find its square root. If you're not familiar with converting repeating decimals, it's worth looking up the method, as it can simplify many similar problems.

6. Calculate √36

Next, we have √36. This is asking us what number, when multiplied by itself, equals 36. The answer is 6, since 6 * 6 = 36. So, √36 = 6. This is another example of a perfect square, and it's crucial to recognize these quickly. Knowing your squares and square roots can significantly speed up your calculations and reduce the chance of errors. If you're struggling with these, practice memorizing the squares of numbers from 1 to 15 – it'll come in handy more often than you think!

7. Calculate √(6^4)

Finally, let's calculate √(6^4). This means we need to find the square root of 6 raised to the power of 4. First, let's calculate 6^4, which means 6 * 6 * 6 * 6 = 1296. Now we need to find the square root of 1296, which is 36. So, √(6^4) = 36. Alternatively, you can recognize that √(6^4) = 6^(4/2) = 6^2 = 36. Understanding exponent rules can simplify these types of problems. By knowing that the square root is equivalent to raising to the power of 1/2, you can easily simplify the expression. This is a more advanced technique, but it can be very useful for more complex problems.

Putting It All Together

Okay, we've simplified each part of the expression. Now, let's plug those values back into the original expression and solve it:

Original expression: √(625/81) : √(25/9) + 3(1/3) : √(1.(7)) - √36 : √(6^4)*

Simplified values:

√(625/81) = 25/9
√(25/9) = 5/3
3*(1/3) = 1
√(1.(7)) = 4/3
√36 = 6
√(6^4) = 36

Substitute these values back into the expression:

(25/9) : (5/3) + 1 : (4/3) - 6 : 36

Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, let's rewrite the divisions as multiplications:

(25/9) * (3/5) + 1 * (3/4) - 6 * (1/36)

Now, let's perform the multiplications:

(25/9) * (3/5) = (25 * 3) / (9 * 5) = 75/45 = 5/3

1 * (3/4) = 3/4

6 * (1/36) = 6/36 = 1/6

So our expression becomes:

5/3 + 3/4 - 1/6

To add and subtract these fractions, we need to find a common denominator. The least common multiple of 3, 4, and 6 is 12. So, let's convert each fraction to have a denominator of 12:

5/3 = (5 * 4) / (3 * 4) = 20/12

3/4 = (3 * 3) / (4 * 3) = 9/12

1/6 = (1 * 2) / (6 * 2) = 2/12

Now we can rewrite the expression as:

20/12 + 9/12 - 2/12

Combine the fractions:

(20 + 9 - 2) / 12 = 27/12

Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

27/12 = (27 ÷ 3) / (12 ÷ 3) = 9/4

So, the final answer is 9/4 or 2.25

Conclusion

Alright, guys, we did it! We successfully solved the radical expression: √(625/81) : √(25/9) + 3(1/3) : √(1.(7)) - √36 : √(6^4)*. Remember, the key is to break down the problem into smaller, manageable steps, simplify each part, and then put it all back together. Keep practicing, and you'll become a radical-solving pro in no time!