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PRINCIPAL ROOTS

38. Since 66+ 36, and (-6)(-6)=+36, it follows that both +6 and 6 are square roots of 36. The two square roots of 36 are usually written in the form ± 6. Similarly the two square roots of a2 are written±a.

The positive square root of a number is called the principal square root. Unless otherwise stated, we shall consider only the principal square root, and write √366. When has ✓ no sign before it, or has the sign before it, the principal square root is always understood.

When we write √ we mean the negative square root.

Thus, √36 or (36) stands for + 6, − √36 or — (36) stands for

± √36 = ± (36)1 = ± 6.

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It can be shown that there are 3 different cube roots of a number, four different fourth roots of a number, and, in general, n different nth roots of a number.

For example, there are three different cube roots of b3; namely, b, − 1 + √ = 3 b2 -1-√-3b. The fractional coefficients in the last two

2

2

roots involve the imaginary number √ 3. Only one of the three roots is real, namely, the root b. We call b the principal root.

In elementary algebra the principal root is the only root usually considered.

I. If a is a positive number, then the principal nth root of a

1

is its positive value; we designate it by a" or Va. When n is an odd number, then this principal root is the only real root.

1

When n is even there is still another real root, namely, — a" or - Va.

Thus, the principal cube root of 64 is (64) cube root of 64 that is real.

or

V/64

= 4; this is the only

The principal fourth root of 81 is (81) or 81

3. There is another

real fourth root of 81; namely, -3, for we see that (-3) (-3) ( − 3) (—3) : 81.

=

II. If a is a negative number, and n is an odd integer, there is no positive root; there is a negative root and that is taken as the principal nth root.

Thus, the principal cube root of 27 is (-27) or

27=- - 3.

III. If a is a negative number, and n is an even integer, then all the roots are imaginary. Imaginary numbers will be discussed more fully later.

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This result may be obtained at once by multiplying the exponents 3 and 4. In this process algebraic expressions are raised to powers; it is called involution.

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It is agreed that each of the symbols a, a, b, Võ, (ab), Wab shall represent only one nth root, namely, the principal

nth root. For the principal real roots the following formulas can be shown to be true:

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Express each of these formulas in words.

The square root of 2 cannot be exactly expressed by the Hindu-Arabic numerals. One can approximate its value by extracting the square root to two, three, or more decimal places, thus: √2 = 1.41*, √2=1.414*, √2=1.4147+, and so on. The radical √2 or 2, and other radicals of the same kind, √3 or 3, √5 or 53, etc., whose values can be found approximately, but not exactly, represent numbers called irrational numbers.

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Find the real fifth root of:

65. 32 25(m+n-p)-5324.

66. 243 a1o(bc)15x-54.

67. What kind of roots of positive numbers have two real values? Which of those two real roots is the principal root?

68. What kind of roots of positive or negative numbers have only one real value?

69. What kind of roots of negative numbers have no real value?

41. Since

SQUARE ROOT

(a + b)2 = a2 + 2 ab + b2,

√a2+2ab+b2 = a + b.

Thus, by inspection, we can extract the square root of a trinomial which is a perfect square.

Or we may find the square root in this way:

Trial divisor,

a2+2ab+b2 a + b

a2

2 a

12 ab + b2

Complete divisor, 2 a + b 2 ab + b2

Thus the square root is a + b.

This simple case will enable us to devise a rule which is applicable to more complicated cases. The procedure is as follows:

I. Extract the square root of the first term and subtract its square from the polynomial, leaving 2 ab + b2.

II. In 2 ab, we see the factor b which we know is the second term of the root. This factor b may be obtained by dividing 2 ab by 2 a. Thus we call 2 a the trial divisor. Since a is the part of the root already found, we see that the trial divisor is double the root already found. After b has been found, add it to the trial divisor and we have 2 ab + b, the complete divisor.

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