2/3, 2√, 1/3, 3√1/2, 4√1/20
To simplify the given expressions, we can simplify any radical and then simplify any fractions.
1. 2/3: This is already in simplest form.
2. 2√: This isn't a fully simplified expression. Please provide the value inside the radical (√).
3. 1/3: This is already in simplest form.
4. 3√1/2: We need to simplify the fraction inside the radical first by writing it as a radical with a rational exponent. The expression becomes 3√(1/2) = 3√(2/4) = 3√(2/2^2) = 3√(2^(1-2)) = 3√(2^(-1)) = 3/(2^(1/3)) = 3/∛2.
5. 4√1/20: We need to simplify the fraction inside the radical first. The expression becomes 4√(1/20) = 4√(1/(2*10)) = 4√(1/(2*2*5)) = 4√(1/(2^2*5)) = 4√(1/(4*5)) = 4√(1/4)√(1/5) = 4(1/2)(√(1/5)) = 2(1/√5) = 2/√5.
Final answers:
1. 2/3
2. Please provide the value inside the radical (√).
3. 1/3
4. 3/∛2
5. 2/√5 (or (2/√5) if you prefer the format)
To simplify the given expressions, we will address each one step-by-step:
1. Simplifying "2/3":
To simplify the fraction 2/3, we can see that it is already in its simplest form. So, the simplified version is 2/3.
2. Simplifying "2√":
It seems that the expression "2√" is incomplete and requires a number or expression to be square rooted. Please provide the appropriate value to continue with the simplification.
3. Simplifying "1/3":
Similar to the first expression, the fraction 1/3 is already simplified. Therefore, 1/3 is the simplified version.
4. Simplifying "3√(1/2)":
To simplify the expression 3√(1/2), we need to find the cube root of 1/2.
The cube root of 1/2 is equal to (1/2)^(1/3).
Thus, 3√(1/2) simplifies to (1/2)^(1/3).
5. Simplifying "4√(1/20)":
Similar to the previous expression, we need to find the fourth root of 1/20.
The fourth root of 1/20 is equal to (1/20)^(1/4).
Thus, 4√(1/20) simplifies to (1/20)^(1/4).
Please note that without specific numerical values, we cannot simplify the expressions any further.
The given list consists of numbers written in simplified radical form. To compare these numbers, it is helpful to convert them into the same format. Let's convert them all into rational numbers.
To convert a radical expression into a rational number, you need to simplify the radical. For square roots (√), you can check if the number under the radical is a perfect square. For example, √4 = 2. Similarly, for cube roots (∛) or higher power roots, you check if the number under the radical is a perfect cube, fourth power, etc.
First, let's simplify the radical expressions:
2/3: No radical symbol, so it's already in rational form.
2√: This expression cannot be simplified further.
1/3: Already in rational form.
3√1/2: The cube root of 1/2 cannot be simplified further because the numerator and denominator are not perfect cubes.
4√1/20: This expression can be simplified. Find the perfect fourth power that divides both the numerator and the denominator. In this case, the fourth root of 1/20 simplifies to 1/√10. Next, rationalize the denominator by multiplying both the numerator and denominator by √10. Thus, it becomes √10/10.
After simplification, the list becomes:
2/3, 2√, 1/3, √10/10, 1/√10
Now, let's compare these numbers:
To compare fractions, we compare their numerators and denominators. Generally, we find a common denominator for all the fractions and then compare their numerators.
However, in this case, since the denominators are already constants, we can easily compare these fractions.
Ordering the list:
1/3 < 2/3
√10/10 < 2√ < 1/√10
So the final order is:
1/3, 2/3, √10/10, 2√, 1/√10