Nowadays modern electronic computers calculate root of a number is not a difficult task.For example, √2704 = 52, it will count all your calculator.Fortunately, the calculator has not only Windows, but also in the normal, even the most simplistic, phone.True if suddenly (a low probability, the calculation of which, incidentally, includes the addition of root), you will find yourself with no available funds, then, alas, have to rely on their brains.
never mind training places.Especially for those who are not often work with numbers, but even more so with the roots.Addition and subtraction of root - a good workout for the mind bored.And I'll show you step by step addition of roots.Examples may include the following expressions.
equation that needs to be simplified:
√2 + 3√48-4 × √27 + √128
This irrational expression.In order to simplify the need to bring it all radicands broad categories.Doing stages:
first number can not be easier.Go to the second term.
3√48 decompose 48 factorization 48 = 2 × 24 or 48 × 16 = 3.The square root of 24 is not an integer, i.e.a fractional remainder.Since we need the exact value, approximate roots are not suitable.The square root of 16 is 4, to make it from the root sign.Get 3 × 4 × √3 = 12 × √3
following expression we have is negative, ie,It is written with a minus -4 × √ (27.) Spread on 27 factors.We get 27 × 3 = 9.We do not use fractional multipliers because of the fractions to calculate the square root of the complex.9 takeaway from the sign, ieWe calculate the square root.The following expression: -4 × 3 × √3 = -12 × √3
√128 next term calculate the part that can be taken out from under the root.128 = 64 × 2, where √64 = 8.If you can imagine it will be easier because this expression: √128 = √ (8 ^ 2 × 2)
Rewriting expression with simplified terms:
√2 + 12 × √3-12 × √3 + 8 × √2
Now we add up the number of the same radicals.You can not add or subtract an expression of different radicals.Addition roots require compliance with this rule.
get the following answer:
√2 + 12√3-12√3 + 8√2 = 9√2
√2 = 1 × √2 - hope that in algebra decided to omit such elements will not benews to you.
expressions may be represented not only the square root, but also with the cubic root or n-th degree.
Addition and subtraction of roots with different exponents, but with equivalent radical expression, as follows:
If we have an expression like √a + ∛b + ∜b, we can simplify this expression as:
∛b + ∜b =12 × √b4 + 12 × √b3
12√b4 + 12 × √b3 = 12 × √b4 + b3
We brought two similar terms to the general terms of the root.Here, it uses the properties of the roots, which states that if the number of degree of radical expression and the number of root index multiplied by the same number, its calculation remains unchanged.
note: the exponents are added only when multiplying.
Consider an example where the expression contains fractions.
5√8-4 × √ (1/4) + √72-4 × √2
We will decide on the steps of:
5√8 = 5 * 2√2 - we make from the root of the retrievable.
- 4√ (1/4) = - 4 √1 / (√4) = - 4 * 1/2 = - 2
If the body is represented by a root fraction, the fraction is not a part of this change, if the square rootof the dividend and divisor.As a result, we have described above equality.
√72-4√2 = √ (36 × 2) - 4√2 = 2√2
10√2 + 2√2-2 = 12√2-2
Here and get the answer.
main thing to remember, that of negative numbers is not extracted from the root of even exponent.If even degree radical expression is negative, the expression is unsolvable.
Adding roots is possible only when the coincidence of the radicals expressions, as they are similar terms.The same applies to the difference.
Addition roots with different numerical exponents carried out by bringing the total extent of the root of both terms.This law has the same effect as a reduction to a common denominator when adding or subtracting fractions.
If there is a radical expression of a number raised to the power of this expression can be simplified by assuming that the root between the index and the extent there is a common denominator.
