Cube Roots & Fourth Roots: Solve A Tricky Math Problem!

Hey guys! Let's dive into a fun math problem involving cube roots and fourth roots. We're given that a = 125^(1/3) and b = 81^(1/4). Our mission, should we choose to accept it, is to determine which of the following statements are true. And remember, there might be more than one correct answer, so keep your eyes peeled!

Breaking Down the Problem

Before we jump into evaluating the statements, let's first figure out the values of a and b. This will make it much easier to determine the truthfulness of each statement.

Calculating the Value of 'a'

We know that a = 125^(1/3). What does this even mean? Well, raising something to the power of 1/3 is the same as finding its cube root. In other words, we're looking for a number that, when multiplied by itself three times, equals 125. Think about it: 5 * 5 * 5 = 125. Therefore, the cube root of 125 is 5. So, a = 5. This is a fundamental concept in mathematics, particularly when dealing with exponents and radicals. Understanding how to compute cube roots (and other roots like square roots and fourth roots) is essential for simplifying expressions and solving equations. The expression 125^(1/3) is simply a compact way of representing the cube root of 125. Remember that the denominator in the exponent represents the index of the radical, and the numerator represents the power to which the base is raised. In this case, since the numerator is 1, we are just taking the cube root of 125. So, any time you see an expression like x^(1/n), think of it as the nth root of x. This understanding will help you tackle more complex problems involving fractional exponents and radicals.

Calculating the Value of 'b'

Now let's tackle b = 81^(1/4). Similar to the cube root, raising something to the power of 1/4 means finding its fourth root. This means we need a number that, when multiplied by itself four times, equals 81. Let's try a few numbers. 1 * 1 * 1 * 1 = 1 (too small). 2 * 2 * 2 * 2 = 16 (still too small). 3 * 3 * 3 * 3 = 81! Bingo! So, the fourth root of 81 is 3. Therefore, b = 3. Just like calculating the cube root, finding the fourth root involves determining what number, when raised to the fourth power, yields the original number. This concept extends to any root, such as the square root (power of 1/2), fifth root (power of 1/5), and so on. The key is to recognize the relationship between the exponent and the root, and to practice identifying perfect powers of numbers. For example, knowing that 81 is a perfect fourth power (3^4) makes it much easier to calculate its fourth root. Similarly, being familiar with perfect squares (4, 9, 16, 25, etc.) and perfect cubes (8, 27, 64, 125, etc.) can greatly speed up your calculations when dealing with roots and exponents.

Evaluating the Statements

Alright, now that we know that a = 5 and b = 3, let's evaluate each statement to see which ones hold true.

Statement 1: Nilai a adalah 25 (The value of a is 25).

This statement is incorrect. We already determined that a = 5, not 25. Easy peasy!

Statement 2: Nilai b adalah 3 (The value of b is 3).

This statement is correct! We calculated that b = 3. Give yourself a pat on the back!

Statement 3: Nilai a setara dengan 8^(1/3) + 27^(1/3) (The value of a is equivalent to 8^(1/3) + 27^(1/3)).

Let's break this down. We need to find the cube root of 8 and the cube root of 27, and then add them together. The cube root of 8 is 2 (since 2 * 2 * 2 = 8). The cube root of 27 is 3 (since 3 * 3 * 3 = 27). So, 8^(1/3) + 27^(1/3) = 2 + 3 = 5. Since a = 5, this statement is also correct! This statement highlights the importance of understanding how roots interact with addition. It's crucial to remember that you can't simply combine the numbers inside the radicals before taking the roots. Instead, you need to calculate each root separately and then perform the addition. This is a common mistake that students often make, so be sure to pay close attention to the order of operations when dealing with expressions involving radicals and addition or subtraction. Also, remember to simplify each radical as much as possible before adding them together. In this case, both 8^(1/3) and 27^(1/3) could be simplified to whole numbers, making the addition straightforward. In other cases, you may need to simplify the radicals by factoring out perfect squares, cubes, or higher powers before you can combine them.

Statement 4: Nilai b setara ... (The value of b is equivalent to ...)

Unfortunately, the fourth statement is incomplete in the original problem. We cannot evaluate its truthfulness without knowing the full statement. However, if we had the complete statement, we would follow the same process: simplify any expressions involving roots, and then compare the result to the value of b (which is 3) to determine if the statement is true or false. For example, let's pretend the statement was: Nilai b setara dengan 16^(1/4) + 1. In this case, 16^(1/4) is 2 (because 222*2 = 16), so 16^(1/4) + 1 = 2 + 1 = 3. Since b=3, this statement would be true. The key to solving these types of problems lies in understanding the definitions of roots and exponents, and applying the correct order of operations. Remember to always simplify radicals as much as possible before performing any addition, subtraction, multiplication, or division. And if you encounter a problem with an incomplete statement, be sure to ask for clarification before attempting to solve it.

Conclusion

So, based on our calculations, the correct statements are:

• Nilai b adalah 3 (The value of b is 3).
• Nilai a setara dengan 8^(1/3) + 27^(1/3) (The value of a is equivalent to 8^(1/3) + 27^(1/3)).

Nice work, guys! You've successfully navigated the world of cube roots and fourth roots. Keep practicing, and you'll become a master of exponents and radicals in no time! Understanding exponents and roots is a cornerstone of algebra and higher-level mathematics. Mastering these concepts will not only help you solve problems in the classroom but also provide you with a solid foundation for tackling more advanced topics such as calculus, differential equations, and linear algebra. So, keep exploring, keep questioning, and keep pushing yourself to learn more. The world of mathematics is vast and fascinating, and there's always something new to discover!