Simplest Radical Form

Summary

This tutorial explains how to simplify radical expressions, starting with square roots. It uses examples such as simplifying the square root of 45 by breaking it into factors, where one factor is a perfect square. The process involves isolating perfect squares, simplifying them, and leaving non-perfect squares under the radical sign. The second example demonstrates this process with the square root of 180, emphasizing the need to break down the numbers further until no perfect square factors remain. The final simplified form is provided with clear explanations of each step.

Takeaways

• 😀 Start by breaking down the number under the radical into two factors, one of which must be a perfect square.
• 😀 For example, to simplify the square root of 45, break it into 9 * 5, where 9 is a perfect square.
• 😀 Write the square roots of each factor separately: square root of 9 and square root of 5.
• 😀 The square root of 9 is 3, so you can write '3' under the radical sign for 9, while the square root of 5 remains under the radical.
• 😀 If a number under the radical does not have a perfect square factor, the expression is already in simplest radical form.
• 😀 For the square root of 180, start by breaking it into 9 * 20, where 9 is a perfect square.
• 😀 The square root of 9 is 3, so write 3 under the radical sign for 9, and keep the square root of 20.
• 😀 Before finalizing the simplification, check if the number under the square root (20) can be broken into smaller perfect squares.
• 😀 20 can be further factored into 4 * 5, where 4 is a perfect square.
• 😀 The square root of 4 is 2, so multiply the results outside the radicals (3 * 2 = 6) and leave the square root of 5 under the radical sign.

Q & A

What is the first step when simplifying the square root of a number like 45?

The first step is to break the number into two factors, ensuring that one of the factors is a perfect square. For 45, the factors are 9 and 5, where 9 is the perfect square.

What do you do once you've broken the number into two factors?

After breaking the number into two factors, you write each factor under its own radical sign. For example, the square root of 9 and the square root of 5 are written separately.

How do you simplify the square root of a perfect square like 9?

The square root of a perfect square, like 9, is simplified to a whole number. For 9, the square root is 3, so you replace the square root of 9 with 3.

What happens if a factor under the radical is not a perfect square?

If a factor under the radical is not a perfect square, like the square root of 5, you leave it as it is and do not attempt to simplify it further.

What is the simplest radical form of the square root of 45?

The simplest radical form of the square root of 45 is 3 times the square root of 5 (3√5).

When simplifying the square root of 180, what is the first step?

The first step is to break 180 into two factors, one of which must be a perfect square. For 180, the factors are 9 and 20, where 9 is a perfect square.

What should you do after separating the square root of 9 and 20 in the square root of 180?

After separating the square root of 9 and 20, you simplify the square root of 9 to 3, and then check if the square root of 20 can be further simplified.

How do you simplify the square root of 20?

You break 20 into two factors: 4 and 5. Since 4 is a perfect square, you can simplify the square root of 4 to 2, leaving the square root of 5 as is.

What is the simplest radical form of the square root of 180?

The simplest radical form of the square root of 180 is 6 times the square root of 5 (6√5).

How do you handle two numbers outside of the radical during simplification?

When you have two numbers outside of the radical, like 3 and 2 in the square root of 180, you multiply them together. In this case, 3 * 2 equals 6.