Back to school. Addition of roots

In our time of modern electronic computers, calculating the root of a number does not seem to be a difficult task. For example, √2704 = 52, this calculates any calculator for you. Fortunately, the calculator is not only in Windows, but also in the usual, even the most simple, phone. True, if suddenly (with a small amount of probability, the calculation of which, among other things, includes the addition of roots), you will find yourself without available means, then, alas, you will only have to rely on your brains.

Training of the mind never puts. Especially for those who do not often work with numbers, much less with roots. Adding and subtracting roots is a good warm-up for a bored mind. And I'll show you step by step the addition of roots. Examples of expressions can be the following.

The equation to be simplified:

√2 + 3√48-4 × √27 + √128

This is an irrational expression. In order to simplify it, we need to bring all the subordinate expressions to the general form. We do in stages:

The first number can not be simplified any more. We pass to the second term.

3: 48 we factor 48 into multipliers: 48 = 2 × 24 or 48 = 3 × 16. The square root of 24 is not an integer; Has a fractional remainder. Since we need exact meaning, the approximate roots do not fit us. The square root of 16 is 4, take it out from under the root sign. We get: 3 × 4 × √3 = 12 × √3

The following expression for us is negative, i.e. Written with a minus sign -4 × √ (27.) We decompose 27 into multipliers. 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 9 from under the sign, i.e. Calculate the square root. We get the following expression: -4 × 3 × √3 = -12 × √3

The next summand √128 computes the part that can be taken out from under the root. 128 = 64 × 2, where √64 = 8. If it is easier for you to represent this expression like this: √128 = √ (8 ^ 2 × 2)

We rewrite the expression with simplified terms:

√2 + 12 × √3-12 × √3 + 8 × √2

Now add up the numbers with the same sub-root expression. You can not add or subtract expressions with different subordinate expressions. Adding roots requires compliance with this rule.

The answer is the following:

√2 + 12√3-12√3 + 8√2 = 9√2

√2 = 1 × √2 - I hope that the fact that it is common in algebra to omit such elements will not become news for you.

Expressions can be represented not only by the square root, but also with the cubic or root of the nth power.

The addition and subtraction of roots with different exponents, but with an equivalent subordinate expression, occurs as follows:

If we have an expression of the form √a + ∛b + ∜b, then we can simplify this expression like this:

∛b + ∜b = 12 × √b4 + 12 × √b3

12√b4 + 12 × √b3 = 12 × √b4 + b3

We brought two similar members to the total root index. Here we used the property of roots, which says: if the number of the degree of the radicand and the number of the root exponent are multiplied by the same number, then its calculation remains unchanged.

Note: the exponents are added only when multiplying.

Consider an example where there are fractions in an expression.

5√8-4 × √ (1/4) + √72-4 × √2

We will decide on the 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 body of the root is represented by a fraction, then often this fraction does not change if the square root of the dividend and divisor is extracted. As a result, we obtained the equality described above.

√72-4√2 = √ (36 × 2) - 4√2 = 2√2

10√2 + 2√2-2 = 12√2-2

So that's the answer.

The main thing to remember is that a root with an even exponent is not extracted from negative numbers. If the even degree of the radicand is negative, then the expression is unsolvable.

The addition of roots is possible only if the subordinate expressions coincide, since they are similar terms. The same applies to the 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 the reduction to a common denominator when adding or subtracting fractions.

If there is a number in the radicand expression raised to a power, then this expression can be simplified, provided that there is a common denominator between the exponent of the root and the degree.