Square Roots of Compound Expressions.

Exercise 78.

Find the square root of: 1. $a^2+2ab+2ac+b^2+2bc+c^2$.

2. $x^4+2x^3+3x^2+2x+1$.

3. $x^4-4x^{3}y+6x^2{y}^2-4x{y}^3+y^4$.

4. $4a^4-12a^{3}b+29a^2{b}^2-30a{b}^3+25b^4$.

5. $16x^6+24x^{5}y+9x^4{y}^2-16x^3{y}^3-12x^2{y}^4+4y^6$.

6. $4x^6-4x^4{y}^2+12x^3{y}^3+x^2{y}^4-6x{y}^5+9y^6$.

181. Arithmetical Square Roots.

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

Since $1=1^2$, $100={10}^2$, $10,000={100}^2$, and so on, it is evident that the square root of a number between $1$ and $100$ lies between $1$ and $10$; of a number between $100$ and $10,000$ lies between $10$ and $100$. In other words, the square root of a number expressed by one or two figures is a number of one figure, of a number expressed by three or four figures is a number of two figures, and so on.

If, therefore, an integral square number is divided into groups of two figures each, from the right to the left, the number of figures in the root will be equal to the number