In our time of modern electronic computers, calculating the root of the number does not seem to be a difficult task. For example, √2704 = 52, any calculator will calculate this for you. Fortunately, the calculator is not only in Windows, but also in a regular, even the most simple, phone. True, if all of a sudden (with a low degree of probability, the calculation of which, by the way, includes the addition of roots) you find yourself without available means, then, alas, you have to rely only on your brains.
Mind training never puts. Especially for those who do not often work with numbers, and even more so with roots. Adding and subtracting roots is a good workout for a bored mind. I’ll also show the addition of the roots in stages. Examples of expressions may be as follows.
The equation to be simplified is:
√2 + 3√48-4 × √27 + √128
This is an irrational expression. In order to simplify it, you need to bring all the radical expressions to a common view. We do it in stages:
The first number cannot be simplified. We pass to the second term.
3√48 we factor 48 into factors: 48 = 2 × 24 or 48 = 3 × 16. The square root of 24 is not integer, i.e. has a fractional remainder. Since we need the exact value, the approximate roots do not suit us. The square root of 16 is 4, take it out from under the sign of the root. We get: 3 × 4 × √3 = 12 × √3
The following expression is negative, i.e. written with a minus sign -4 × √ (27.) Factor 27 into factors. We get 27 = 3 × 9. We do not use fractional factors, because it is more difficult to calculate the square root of fractions. We take out 9 from under the sign, i.e. calculate the square root. We get the following expression: -4 × 3 × √3 = -12 × √3
The next term √128 is the part that can be taken out from under the root. 128 = 64 × 2, where √64 = 8. If it will be easier for you, you can imagine this expression as follows: √128 = √ (8 ^ 2 × 2)
Rewrite the expression with simplified terms:
√2 + 12 × √3-12 × √3 + 8 × √2
Now add up the numbers with the same radical expression. You cannot add or subtract expressions with different root expressions. Addition of roots requires compliance with this rule.
The answer is as follows:
√2 + 12√3-12√3 + 8√2 = 9√2
√2 = 1 × √2 - I hope that the fact that it is customary to omit such elements in algebra will not be news to you.
Expressions can be represented not only by a square root, but also with a cubic or root of the nth degree.
Addition and subtraction of roots with different exponents, but with an equivalent radical expression, occurs as follows:
If we have an expression of the form √a + ∛b + ∜b, then we can simplify this expression as follows:
∛b + ∜b = 12 × √b4 + 12 × √b3
12√b4 + 12 × √b3 = 12 × √b4 + b3
We have brought two similar members to a common measure of root. Here we used the property of roots, which states: if the number of the degree of the radical expression and the number of the root exponent are multiplied by the same number, then its calculation will remain unchanged.
Note: exponents add up only when multiplied.
Consider an example where fractions are present in an expression.
5√8-4 × √ (1/4) + √72-4 × √2
We will decide by stages:
5√8 = 5 * 2√2 - we take out the extracted part from under the root.
- 4√ (1/4) = - 4 √1 / (√4) = - 4 * 1/2 = - 2
If the root is represented by a fraction in the body, then often this fraction does not change if we extract the square root of the dividend and divisor. As a result, we have the equality described above.
√72-4√2 = √ (36 × 2) - 4√2 = 2√2
10√2 + 2√2-2 = 12√2-2
That is the answer.
The main thing to remember is that a root with an even exponent is not extracted from negative numbers. If an even degree radical expression is negative, then the expression is unsolvable.
Addition of roots is possible only if the radical expressions coincide, since they are similar terms. The same applies to difference.
The addition of roots with different numerical exponents is done by bringing both terms to the common root degree. This law acts in the same way as reducing to a common denominator when adding or subtracting fractions.
If in the root expression there is a number raised to a power, then this expression can be simplified provided that there is a common denominator between the root and degree.