Involution of Radicals
283. Radical quantities, like powers, are involved by multiplying the index of the root into the index of the required power.
1. The square of a1/3 =a(1/3).2 = a2/3. Fora1/3a1/3 = a2/3.
2. The cube of a1/4 = a(1/4).3 = a3/4. For a1/4.a1/4 = a2/3.
3. And universally, the n-th power of a1/m = a(1/m).n = an/m.
For the n-th power, of a1/m = 1/m.a1/m... n times, and the sum of the indices will then be n/m.
4. The square of a2/3x3/4, is a4/3x6/4.
284. A root is raised to a power of the same name, by removing the index of the radical sign.
Thus the cube of 3√b + x, is b + x.
And the n-th power of (a - y)1/n, is (a - y).
285. When the radical quantities have rational coefficients, these must also be involved.
1. The square ofan√x is a2n√ .
2. The cube of 3a3√y, is 27a3y.
286. But if the radical quantities are connected with others by the signs + and -, they must be involved by a multiplication of the several terms, as in Art. 209.
Ex. 1. Required the squares of a + √y and a - √y.
(a + √y)(a + √y) = a2 + 2a√y + √y.√y = a2 + 2a√y + √ = a2 + 2a√y + y
(a - √y)(a - √y) = a2 - 2a√y + √y.√y = a2 - 2a√y + √ = a2 - 2a√y + y.
2. Required the cube of a - √b.
3. Required the cube of 2d + √x.
287. It is unnecessary to give a separate rule for the evolution of radical quantities, that is, for finding the root of a quantity which is already a root. The operation is the same as in other cases of evolution. The fractional index of the radical quantity is to be divided, by the number expressing the root to be found. Or, the radical sign belonging to the required root, may be placed over the given quantity. (Art. 252.) If there are rational coefficients, the roots of these must also be extracted.
Thus, the square root of a1/3, is a(1/3):2 = a1/6.
The cube root of a(xy)1/2 is a1/3(xy)1/6.
288. It may be proper to observe, that dividing the fractional index of a root is the same in effect, as multiplying the number which is placed over the radical sign. For this number corresponds with the denominator of the fractional index; and a fraction is divided, by multiplying its denominator
Thus √a = a1/2. 6√a = a1/6.
4√a = a1/4. 2n√a = a1/2n.
On the other hand, multiplying the fractional index is equivalent to dividing the number which is placed over the radical sign.
Thus the square of 6√a or a1/6, is 3√a or a(1/6).2 = a1/3.
288. b. In algebraic calculations, we have sometimes occasion to seek for a factor, which multiplied into a given radical quantity, will render the product rational. In the case of a simple radical, such a factor is easily found. For if the n-th root of any quantity, be multiplied by the same root raised to a power whose index is n - 1, the product will be the given quantity.
Thus n√x.n√ or x1/n.x(n-1)/n = xn/n =x.
And (x + y)1/n.(x + y)(n-1)/n = x + y.
And4√a.4√ = a.
288. c. A factor which will produce a rational product, when multiplied into a binomial surd containing only the square root, may be found by applying the principle, that the product of the sum and difference of two quantities, is equal to the difference of their squares. (Art. 231.) The binomial itself, after the sign which connects the terms is changed from + to -, or from - to +, will be the factor required.
Thus (√a + √b).(√a - √b) = (√a - √b) = √ - √ = a - b, which is free from radicals.
So (1 + √2)(1 - √2) = 1 - 2 = -1.
And (3 - 2√2)(3 + 2√2) = 1.
When the compound surd consists of more than too terms, it may be reduced, by successive multiplications, first to a binomial surd, and then to a rational quantity.
Thus (√10 - √2 - √3)(√10 + √2 + √3) = 5 - 2√6 a binomial surd.
And (5 - 2√6).(5 + 2√6) = 1.
Therefore (√10 - √2 - √3) multiplied into (√10 - √2 + √3).(5 + 2√6) = 1.
288. d. It is sometimes desirable to clear from radical signs the numerator or denominator of a fraction. This may be effected, without altering the value of the fraction, if the numerator and denominator be both multiplied by a factor which will render either of them rational, as the case may require.
1. If both parts of the fraction √a/√x multiplied by √a, it will become √a.√a/√x.√a = a/√ax in which the numerator is a rational quantity. Or if both parts of the given fraction be multiplied by √x it will become √ax/x, in which the denominator is rational.
2. The fraction
3. The fraction
4. The fraction 6/51/4 = 6.53/4/51/4+3/4 = 64√125/5.
5. Reduce 2/√3 to a fraction having a rational denominator.
6. Reduce (a - √b)/(a + √b) to a fraction having a rational denominator.
288. e. The arithmetical operation of finding the proximate value of a fractional surd, may be shortened, by rendering either the numerator or the denominator rational. The root of a fraction is equal to the root of the numerator divided by the root of the denominator. (Art. 255.)
When the fraction is thrown into this form, the process of extracting the root arithmetically, will be confined either to the numerator, or to the denominator.
Thus the square root of 3/7 = √3/√7 = √3.√7/√7.√7 = √21/7.
Examples for practice.
1. Find the 4th root of 81.
2. Find the 6th root of (a + b)-3.
3. Find the n-th root of (x - y)1/6.
4. Find the cube root of -125ax6.
5. Find the square root of 4a4/9x2y2.
6. Find the 5th root of 32a5x10/243.
7. Find the square root of x2 - 6bx + 9b2.
8. Find the square root of a2 + ay + y2/4.
9 Reduce ax2 to the form of the 6th root.
10. Reduce -3y to the form of the cube root.
11. Reduce a2 and a1/3 to a common index.
12. Reduce 41/3 and 51/4 to a common index.
13. Reduce a1/2 and b1/4 to the common index 1/5.
14. Reduce 21/2 and 41/4 to the common index 1/3.