Contents
In this publication, we will consider how to take out numbers (multipliers) and letters from under the sign of the root of the second and higher degrees. The information is accompanied by practical examples for better understanding.
Rule of removal from under the root
Square root
Take out the number (multiplier) from under the root sign – this means extracting the root from the radical expression (that is, what is under the sign of the root).
If a2 = b, then √b = a.
For example:
- √4 = 2, because 22 = 4;
- √36 = 6, because 62 = 36.
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nth root
To remove the root expression from under the root of the third and higher degrees, we extract the root to the appropriate degree.
For example:
Example 1
Take out the factor from under the root
Decision:
In this case, you can extract the square root only from the number twenty-five, which we will do.
Example 2
Take out the factor from √45.
Decision:
1. First, we decompose the radical expression (the number 45) into factors. In our case, these are 9 and 5.
2. From the obtained numbers, you can extract the square root only from nine. Thus we get:
Valid actions under the root
If it is required to remove an expression from under the root, then this can be done only in relation to the work.
For example:
- √16 · 5 = √16 · √5 (right)
- √25 + 11 ≠ √25 +√11 (not properly)
- √47 – 38 ≠ √47 – √38 (not properly)
- √8:2 ≠ √8 : √2 (not properly)
With the exception of the first option, in other cases, you must first perform actions under the root, and then extract it.
- √25 + 11 = √36 = 6
- √47 – 38 = √9 = 3
- √8:2 = √4 = 2
Making a letter
Take the letter out from under the root – this is the same as raising it to a fraction, where the numerator is the degree of the radical expression, and the denominator is the root itself.
Note: the same formula can be used by substituting specific numbers instead of a letter.
For example:
