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And y=a+2b√n—✅a—2b↓n


Which are general expressions for the values of a and y.

If now, we take n=-1, or x2-y-a, the above expressions become,

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These values, though expressed by imaginary quantities are real ones. For putting -1 instead of n, in equations M, N, and multiplying, we have

x2+y2=✅a2+4b3, (P.)

But from the first equation, x3—y2=d.

Consequently, by addition and subtraction,

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The same conclusions may be obtained by squaring the values of x and y first found, and extracting the square roots of the results.

The impossible surd -1, was evidently introduced into the result, by adopting a process in the general solution, which was not applicable to the particular equation x-y-a; yet the values of x and y, when cleared of imaginary surds, are the true ones; as may be shown by a different solution of the problem.

To the square of the first equation adding 4 times the square of the second, x+2x y2+y2=a2+4bo.

Whence by evolution, x+ya+46, the same as equation P, deduced from imaginary surds.

106. From what is shown in the foregoing article, we readily infer, that when, in the solution of a problem, the value of the quantity sought appears in terms of imagin ary surds, we are not thence immediately to conclude, that the data are inconsistent; as the adoption of an inapplicable process may produce such a result, yet in this case the imaginary surd may be eliminated by the use of proper expedients. When, however, the data are inconsistent, no analytical address can clear the final equation of its impossible quantities.

Imaginary surds differ from real ones in this important particular. Real surds, however complex, admit of an approximation to their value; but imaginary surds admit of no approximation, and must either be eliminated, when practicable, or remain the intractable indications of incongruous assumptions.

The following cases exhibit the most useful applications of algebra to surd quantities.


107. To reduce a rational quantity to the form of a surd.

Raise the given quantity to the power denoted by the index of the surd, and to this power apply the radical sign or index proposed.


1. Reduce 5 to the form of a square root, and a to that of a 4th root.

5=√✓5a=√✓/25, and a=(a4)—

2. Reduce 3 to the form of a cube root.

Result, 27.


3. Express—a in the form of a cube root.

Result, (———279)

4. Reduce 2/5 to the form of a square root.

Result, 20.

5. Reduce 3/2 to the form of a 4th root.

Result, (324)3.

6. Express a+b in the form of a square root.

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108. To reduce radical quantities, having different indices, to other equivalent quantities with a common radical sign.

Reduce the fractional exponents to a common denominator, involve the given quantities to the powers denoted by their respective numerators; and to the results apply the reciprocal of the common denominator as the common exponent.



1. Reduce a, b, and c to equivalent quantities, having a common exponent.

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2. Reduce 3, and 4 to a common index.

Result, 27, 16

3. Reduce a3, b, to a common radical sign.

Result, ais, bis

4. Reduce (a+x)3, (a) to a common index. Result, (a2+2ax+x3)3, and (a3—3a2x+3ax2—v3)*. 5. Reduce a3, and a to the common index 1*


Result, (a1)12, (23) 12.

6. Reduce 43, and 52

to a common index.

Result, (2564), and 254.


109. To reduce surds to their most simple terms.

Resolve the surd, if possible, into two factors, one of which shall be the greatest power it contains. Extract the root, and thereto aunex the other factor with its proper radical sign.


1. Reduce 250 to its simplest terms.

250-125×2=53×2. .. 3/250=532.

2. Reduce 32 to its simplest terms.

Result, 4/2.

* When the common index, to which the fractional exponents are to be reduced is given, divide each of the given exponents by that common index, and involve the given quantities to the powers indicated by the quotients.

3. Reduce 243 to its simplest form.

4. Reduce 5184 to simple terms.

5. Reduce (144) to its simplest terms.

6. Reduce (a3b—a3x)3 to simple terms.

Result, 34/3.

Result, 123.

Result, 2/3.

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110. To add or subtract surd quantities.

Reduce the quantities, when fractional, to a common denominator; and, when the exponents are different, to a common radical sign. Also express the surds in their simplest terms. If then the reduced surds are alike, they may be added or subtracted as other algebraic quan

* When the given surd is fractional, the denominator may generally be made rational, without changing the value of the fraction, by multiplying both terms by proper numbers or quantities. When the denominator consists of two quadratic surds connected by the sign + or, the proper multiplier consists of the same surds connected by the opposite sign.


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