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Theorie

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Square roots

The well-known symbol a , the square root of a, is used to describe the number that when multiplied by itself gives a. However, one has to be a little more precise in defining this symbol.

The equation x2=4 has two solutions x=2 and x=−2, since both 22=4 and (−2)(−2)=4. It would then be logical to suppose that 4  can be either −2 or 2, i.e. 4=2 , but 4  only denotes the positive number 2.

It is therefore wrong to state that 4=2  but correct to state that the equation x2=4 has the solution x=2.

When taking square roots, it is useful to know some methods of calculation. As a=a12  we can use the laws of exponents as "laws of roots". For example, we have

94=(94)12=912412=94.

In this way we obtain the following rules for quadratic roots, which apply to all real numbers ab0:

( We must however, in the above division, assume as always that b is not 0.)

Note that the above calculations assume that a and b0. If a and b are negative (< 0) then a  and b  are not defined as real numbers. It is tempting to write , for example,

−1=−1−1=(−1)(−1)=1=1

but something here cannot be right. The explanation is that −1  is not a real number, which means the laws of roots discussed above may not be used.

Higher order roots

The cube root of a number a is defined as the number that multiplied by itself three times gives a, and is denoted as 3a .

Note that, unlike square roots, cube roots are also defined for negative numbers.

For any positive integers n one can define the n'th root of a number a as

• if n is even and a0 then na  is the non-negative number that when multiplied by itself n times gives a,
• if n is odd, na  is the number that when multiplied by itself n times gives a.

The root na  can also be written as a1n .

For n'th roots the same rules apply as for quadratic roots if ab0. Note that if n is odd these methods apply even for negative a and b, that is, for all real numbers a and b.

Simplification of expressions containing roots

Often, one can significantly simplify expressions containing roots by using the usual methods for roots . As is also the case when using the laws of exponents, it is desirable to reduce expressions into as "small" roots as possible. For example, it is a good idea to do the following

8=42=42=22

because it helps simplification as we see here

28=222=2.

By rewriting expressions containing roots in terms of "small" roots one can also sum roots of "the same kind", e.g.

8+2=22+2=(2+1)2=32.

Rational root expressions

When roots appear in a rational expression one often wants to avoid roots in the denominator (because it is difficult with hand calculations to divide by irrational numbers). By multiplying the numerator and denominator by 2  for example, one obtains

12=1222=22

which usually is preferable.

In other cases, you can take advantage of the difference of two squares method, (a+b)(a−b)=a2–b2. One multiplies the numerator and denominator by the denominator´s “conjugate” expression and the root sign is eliminated from the denominator by squaring, as in the following,

32+1=32+12−12−1=3(2−1)(2+1)(2−1)=(2)2−1232−31=2−132−3=16−3=6−3.

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