# 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 a^{1/3} =a^{(1/3).2} = a^{2/3}. Fora^{1/3}a^{1/3} = a^{2/3}.

2. The cube of a^{1/4} = a^{(1/4).3} = a^{3/4}. For a^{1/4}.a^{1/4} = a^{2/3}.

3. And universally, the n-th power of a^{1/m} = a^{(1/m).n} = a^{n/m}.

For the n-th power, of a^{1/m} = ^{1/m}.a^{1/m}... n times, and the sum of the indices will then be n/m.

4. The square of a^{2/3}x^{3/4}, is a^{4/3}x^{6/4}.

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 ofa^{n}√x is a^{2}^{n}√x^{2}.

2. The cube of 3a^{3}√y, is 27a^{3}y.

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) = a^{2} + 2a√y + √y.√y = a^{2} + 2a√y + √y^{2} = a^{2} + 2a√y + y

and

(a - √y)(a - √y) = a^{2} - 2a√y + √y.√y = a^{2} - 2a√y + √y^{2} = a^{2} - 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 a^{1/3}, is a^{(1/3):2} = a^{1/6}.

The cube root of a(xy)^{1/2} is a^{1/3}(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 = a^{1/2}. ^{6}√a = a^{1/6}.

^{4}√a = a^{1/4}. ^{2n}√a = a^{1/2n}.

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 a^{1/6}, is ^{3}√a or a^{(1/6).2} = a^{1/3}.

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}√x^{n-1} or x^{1/n}.x^{(n-1)/n} = x^{n/n} =x.

And (x + y)^{1/n}.(x + y)^{(n-1)/n} = x + y.

And^{4}√a.^{4}√a^{3} = 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^{2} - √b^{2} = 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 6/5^{1/4} = 6.5^{3/4}/5^{1/4+3/4} = 6^{4}√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 -125ax^{6}.

5. Find the square root of 4a^{4}/9x^{2}y^{2}.

6. Find the 5th root of 32a^{5}x^{10}/243.

7. Find the square root of x^{2} - 6bx + 9b^{2}.

8. Find the square root of a^{2} + ay + y^{2}/4.

9 Reduce ax^{2} to the form of the 6th root.

10. Reduce -3y to the form of the cube root.

11. Reduce a^{2} and a^{1/3} to a common index.

12. Reduce 4^{1/3} and 5^{1/4} to a common index.

13. Reduce a^{1/2} and b^{1/4} to the common index 1/5.

14. Reduce 2^{1/2} and 4^{1/4} to the common index 1/3.