Cube Roots of Compound Expressions.

The next term of the root is found by dividing $- 3 x^{5}$ , the first term of the remainder after $x^{6}$ is subtracted, by $3 x^{4}$ , and the first complete divisor, $3 x^{4} - 3 x^{3} + x^{2}$ , is found by annexing to the trial divisor $(3 x^{2} - x) (- x)$ , which expression corresponds to $(3 a + b) b$ in the typical form.

The part of the root already found ( $a$ ) is now represented by $x^{2} - x$ , therefore $3 a^{2}$ is represented by $3 (x^{2} - x)^{2} = 3 x^{4} - 6 x^{3} + 3 x^{2}$ , the second trial divisor, and $(3 a + b) b$ by $(3 x^{2} - 3 x - 1) (- 1)$ , since $b$ in this case is found to be $- 1$ , therefore, in the second complete divisor, $3 a^{2} + (3 a + b) b$ is represented by $(3 x^{4} - 6 x^{3} + 3 x^{2}) + (3 x^{2} - 3 x - 1) (- 1) = 3 x^{4} - 6 x^{3} + 3 x + 1 .$

Exercise 80.

Find the cube root of

$x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ .

$8 x^{3} - 12 x^{2} + 6 x - 1$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3}$ .

$64 a^{3} - 144 a^{2} x + 108 a x^{2} - 27 x^{3}$ .

$1 + 3 x + 6 x^{2} + 7 x^{3} + 6 x^{4} + 3 x^{5} + x^{6}$ .

$x^{6} - 3 x^{5} + 6 x^{4} - 7 x^{3} + 6 x^{2} - 3 x + 1$ .

187. Arithmetical Cube Roots.

In extracting the cube root of a number expressed by figures, the first step is to mark it off into groups.

Since $1 = 1^{3}$ , $1000 = 10^{3}$ , $1, 000, 000 = 100^{3}$ , and so on, it follows that the cube root of any number between $1$ and $1000$ , that is, of any number which has one, two, or three figures, is a number of one figure, and that the cube root of any number between $1000$ and $1, 000, 000$ , that is, of any number which has four, five, or six figures, is a number of two figures, and so on.

If, therefore, an integral cube number be divided into groups of three figures each, from right to left, the number of figures in the root will be equal to the number of groups. The last group to the left may consist of one, two, or three figures.

If the cube root of a number have decimal places, the number itself will have three times as many. Thus, if $0.11$ be the cube root of a number, the number is $0.11 \times 0.11 x 0.11 = 0.001331$ . Hence, if a given number contain a decimal, we divide the figures of the number into groups of three figures each, by beginning at the decimal point and marking toward the left for the integral number, and

toward the right for the decimal. We must be careful to have the last group on the right of the decimal point contain three figures, annexing ciphers when necessary.

Extract the cube root of $42875$ .

$r r r l l l & & & 42 & 875 & (35 & & a^{3} = & 27 \cline 4 - 5 3 a^{2} = & 3 \times 30^{2} = & \multicolumn 1 @ r | 2700 & 15 & 875 3 a b = & 3 \times (30 \times 5) = & \multicolumn 1 @ r | 450 b^{2} = & 5^{2} = & \multicolumn 1 @ r | 25 \cline 3 - 3 & & \multicolumn 1 @ r | 3175 & 15 & 875 \cline 4 - 5$

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