Algebra Chapter 10 Simplification Of Fractions
Introduction
In arithmetic, you have learned a lot about fractions. By 5/7 part of an article we mean any 5 parts of the article when it is divided into 7 equal parts. Also, you know that for the fraction 5/7, 5 is the numerator and 7 is the denominator. The conception of fractions in algebra is similar to that in arithmetic.
The general form of a fraction in algebra
In algebra, we denote a fraction by a/b. It represents a fraction for any value of a and b (except b = 0). a/b may also be written saves and it indicates any a parts of a quantity when it is divided into b equal parts. In any fraction, the alphabetic symbol above the fraction line is the numerator and the alphabetic symbol below the line is the denominator. So in the fraction a/b, a is the numerator and b is the denominator.
If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the value of the fraction remains unchanged.
Thus, a/b = a x x / b x x and a/b = a ÷ x / b ÷ x
The sign of a fraction is positive when both the numerator and denominator are of the same sign (i.e., either both + or both -).
The sign of a fraction is negative when the numerator and denominator are of the opposite sign (i.e., one is + and the other is -).
The Lowest form of a fraction
Like arithmetic, an algebraic fraction can also be reduced to its lowest form.
An algebraic fraction is said to be in its lowest form when there does not exist any factor common to both the numerator and denominator.
Like arithmetic, we can reduce a fraction to its lowest form by dividing both the numerator and denominator by their H.C.F.
For Example : \(\frac{x^2 y}{x y^2}=\frac{x^2 y \div x y}{x y^2 \div x y}=\frac{x}{y}\)
In practice, to reduce a fraction into its lowest form we resolve both the numerator and denominator of the fraction into factors and then their common factors are canceled. Then the given fraction reduces to its lowest form.
Read And Learn More WBBSE Solutions For Class 8 Maths
Algebra Chapter 10 Simplification Of Fractions Some examples
Example 1
Reduce \(\frac{18 x^5 y^6 z^9}{24 x^2 y^3 z^8}\) into its lowest term.
Solution:
Given \(\frac{18 x^5 y^6 z^9}{24 x^2 y^3 z^8}\).
= \(\frac{18 x^5 y^6 z^9}{24 x^2 y^3 z^8}=\frac{6 \times 3 \times x^5 \times y^6 \times z^9}{6 \times 4 \times x^2 \times y^3 \times z^8}\)
= \(\frac{3 \times x^3 \times y^3 \times z}{4}=\frac{3 x^3 y^3 z}{4}\)
Example 2
Reduce \(\frac{56 a^5 b c^2}{42 a^2 b^3 c^4}\) into its lowest term.
Solution:
Given \(\frac{56 a^5 b c^2}{42 a^2 b^3 c^4}\).
\(\frac{56 a^5 b c^2}{42 a^2 b^3 c^4}=\frac{14 \times 4 \times a^5 \times b \times c^2}{14 \times 3 \times a^2 \times b^3 \times c^4}\)= \(\frac{4 \times a^4}{3 \times b^2 \times c^2}=\frac{4 a^3}{3 b^2 c^2}\)
= \(\frac{4 a^3}{3 b^2 c^2}\)
WBBSE Class 8 Simplification of Fractions Notes
Example 3
Reduce \(\frac{x^2+3 x+2}{x^2+4 x+3}\) into its lowest term.
Solution:
Given \(\frac{x^2+3 x+2}{x^2+4 x+3}\).
\(\frac{x^2+3 x+2}{x^2+4 x+3}=\frac{x^2+2 x+x+2}{x^2+3 x+x+3}\)= \(\frac{x(x+2)+1(x+2)}{x(x+3)+1(x+3)}=\frac{(x+2)(x+1)}{(x+3)(x+1)}=\frac{x+2}{x+3}\)
= \(\frac{x+2}{x+3}\)
Example 4
Reduce \(\frac{25 x^2-36 y^2}{5 x-6 y}\) into its lowest term.
Solution:
Given \(\frac{25 x^2-36 y^2}{5 x-6 y}\).
\(\frac{25 x^2-36 y^2}{5 x-6 y}=\frac{(5 x)^2-(6 y)^2}{5 x-6 y}\)= \(\frac{(5 x+6 y)(5 x-6 y)}{5 x-6 y}=5 x+6 y\)
= 5x + 6y
Example 5
Reduce \(\frac{15\left(a^3-b^3\right)}{25\left(a^2+a b+b^2\right)}\) into its lowest term.
Solution:
Given \(\frac{15\left(a^3-b^3\right)}{25\left(a^2+a b+b^2\right)}\)
\(\frac{15\left(a^3-b^3\right)}{25\left(a^2+a b+b^2\right)}\)= \(\frac{5 \times 3 \times(a-b)\left(a^2+a b+b^2\right)}{5 \times 5 \times\left(a^2+a b+b^2\right)}=\frac{3(a-b)}{5}\)
= \(\frac{3(a-b)}{5}\)
Example 6
Reduce \(\frac{\left(x^3+y^3\right)\left(x^3-y^3\right)}{x^4+x^2 y^2+y^4}\) into its lowest term.
Solution:
Given
\(\frac{\left(x^3+y^3\right)\left(x^3-y^3\right)}{x^4+x^2 y^2+y^4}\)= \(\frac{\left(x^3+y^3\right)\left(x^3-y^3\right)}{x^4+x^2 y^2+y^4}\)
= \(\frac{(x+y)\left(x^2-x y+y^2\right)(x-y)\left(x^2+x y+y^2\right)}{\left(x^2\right)^2+2 \times x^2 \times y^2+\left(y^2\right)^2-x^2 y^2}\)
= \(\frac{(x+y)(x-y)\left(x^2-x y+y^2\right)\left(x^2+x y+y^2\right)}{\left(x^2+y^2\right)^2-(x y)^2}\)
= \(\frac{(x+y)(x-y)\left(x^2-x y+y^2\right)\left(x^2+x y+y^2\right)}{\left(x^2+y^2+x y\right)\left(x^2+y^2-x y\right)}\)
= \((x+y)(x-y)=x^2-y^2\)
= \(x^2-y^2\)
Step-by-Step Guide to Simplifying Fractions
Application of four basic operations on fractions
Addition, subtraction, multiplication, and division of fractions are similar to those in arithmetic.
For Example:
1. \(\frac{\mathrm{a}}{\mathrm{b}}+\frac{\mathrm{c}}{\mathrm{d}}=\frac{\mathrm{ad}+\mathrm{bc}}{\mathrm{bd}}\)
2. \(\frac{\mathrm{p}}{\mathrm{q}}-\frac{\mathrm{r}}{\mathrm{s}}=\frac{\mathrm{ps}-\mathrm{qr}}{\mathrm{qs}}\)
3. \(\frac{x}{y} \times \frac{p}{q}=\frac{p x}{q y}\)
4. \(\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}\)
= \(\frac{\mathrm{ad}}{\mathrm{bc}}\)
Some Examples
Example 1
Simplify: \(\frac{9 x^2-16 y^2}{x^2-16} \times \frac{x^2-4 x}{3 x-4 y}\)
Solution:
Given \(\frac{9 x^2-16 y^2}{x^2-16} \times \frac{x^2-4 x}{3 x-4 y}\)
= \frac{9 x^2-16 y^2}{x^2-16} \times \frac{x^2-4 x}{3 x-4 y}
= \frac{(3 x)^2-(4 y)^2}{(x)^2-(4)^2} \times \frac{x(x-4)}{3 x-4 y}
= \frac{(3 x+4 y)(3 x-4 y)}{(x+4)(x-4)} \times \frac{x(x-4)}{3 x-4 y}
= \frac{x(3 x+4 y)}{x+4}
Example 2
Simplify: \(\frac{x^3-y^3}{x+y} \times \frac{x^2-y^2}{x^2+x y+y^2}\)
Solution:
Given \(\frac{x^3-y^3}{x+y} \times \frac{x^2-y^2}{x^2+x y+y^2}\).
= \(\frac{x^3-y^3}{x+y} \times \frac{x^2-y^2}{x^2+x y+y^2}\)
= \(\frac{(x-y)\left(x^2+x y+y^2\right)}{x+y} \times \frac{(x+y)(x-y)}{x^2+x y+y^2}\)
= \((x-y)^2\)
Example 3
Simplify: \(\frac{a}{a-b}+\frac{b}{a+b}+\frac{2 a b}{b^2-a^2} .\)
Solution:
Given \(\frac{a}{a-b}+\frac{b}{a+b}+\frac{2 a b}{b^2-a^2} .\).
= \(\frac{\mathrm{a}}{\mathrm{a}-\mathrm{b}}+\frac{\mathrm{b}}{\mathrm{a}+\mathrm{b}}+\frac{2 \mathrm{ab}}{\mathrm{b}^2-\mathrm{a}^2}\)
= \(\frac{a}{a-b}+\frac{b}{a+b}-\frac{2 a b}{a^2-b^2}\)
= \(\frac{a}{a-b}+\frac{b}{a+b}-\frac{2 a b}{(a+b)(a-b)}\)
= \(\frac{a(a+b)+b(a-b)-2 a b}{(a-b)(a+b)}\)
= \(\frac{a^2+a b+a b-b^2-2 a b}{(a-b)(a+b)}=\frac{a^2-b^2}{a^2-b^2}=1\)
Example 4
Simplify: \(\frac{1}{a^2-8 a+15}+\frac{1}{a^2-4 a+3}-\frac{2}{a^2-6 a+5}\)
Solution:
Given \(\frac{1}{a^2-8 a+15}+\frac{1}{a^2-4 a+3}-\frac{2}{a^2-6 a+5}\)
= a2 – 8a + 15
= a2 – 5a – 3a + 15
= a(a – 5) – 3 (a – 5)
= (a – 5) (a – 3) a2 – 4a + 3
= a2 – 3a – a + 3
= a(a – 3) – 1(a – 3)
= (a – 3) (a – 1) a2 – 6a + 5
= a2 – 5a – a + 5
= a(a – 5) – 1(a – 5)
= (a – 5) (a – 1)
Hence, the given expression
\(\frac{1}{a^2-8 a+15}+\frac{1}{a^2-4 a+3}-\frac{2}{a^2-6 a+5}\) = (a – 5) (a – 1)
Understanding Equivalent Fractions
Example 5
Simplify: \(\frac{8 a^3}{a^2+a b+b^2} \times \frac{a+b}{2 a\left(a^3+b^3\right)} \times \frac{a^4+a^2 b^2+b^4}{4 a^2} .\)
Solution:
Given \(\frac{8 a^3}{a^2+a b+b^2} \times \frac{a+b}{2 a\left(a^3+b^3\right)} \times \frac{a^4+a^2 b^2+b^4}{4 a^2} .\)
= \(\frac{8 a^3}{a^2+a b+b^2} \times \frac{a+b}{2 a\left(a^3+b^3\right)} \times \frac{a^4+a^2 b^2+b^4}{4 a^2}\)
= \(\frac{8 a^3}{a^2+a b+b^2} \times \frac{a+b}{2 a(a+b)\left(a^2-a b+b^2\right)}\times \frac{a^4+a^2 b^2+b^4}{4 a^2}\)
= \(\frac{8 a^3(a+b)\left(a^4+a^2 b^2+b^4\right)}{8 a^3(a+b)\left(a^4+a^2 b^2+b^4\right)}=1\)
Example: 6
Simplify: \(\frac{x-y}{x y}+\frac{y-z}{y z}+\frac{z-x}{z x} .\)
Solution:
Given \(\frac{x-y}{x y}+\frac{y-z}{y z}+\frac{z-x}{z x} .\).
= \(\frac{x-y}{x y}+\frac{y-z}{y z}+\frac{z-x}{z x}\)
= \(\frac{x}{x y}-\frac{y}{x y}+\frac{y}{y z}-\frac{z}{y z}+\frac{z}{z x}-\frac{x}{z x}\)
= \(\frac{1}{y}-\frac{1}{x}+\frac{1}{z}-\frac{1}{y}+\frac{1}{x}-\frac{1}{z}=0\)
Example 7
Simplify: \(\frac{x-y}{x(x+y)} \div \frac{x^2+y^2}{2 x^2} \times \frac{2\left(x^4-y^4\right)}{x^2-2 x y+y^2}\)
Solution:
Given \(\frac{x-y}{x(x+y)} \div \frac{x^2+y^2}{2 x^2} \times \frac{2\left(x^4-y^4\right)}{x^2-2 x y+y^2}\)
= \(\frac{x-y}{x(x+y)} \div \frac{x^2+y^2}{2 x^2} \times \frac{2\left(x^4-y^4\right)}{x^2-2 x y+y^2}\)
= \(\frac{x-y}{x(x+y)} \times \frac{2 x^2}{x^2+y^2} \times \frac{2\left(x^4-y^4\right)}{x^2-2 x y+y^2}\)
= \(\frac{x-y}{x(x+y)} \times \frac{2 x^2}{x^2+y^2} \times \frac{2\left(x^2+y^2\right)(x+y)(x-y)}{(x-y)^2}\)
= 4x
Common Mistakes in Simplifying Fractions
Example 8
Simplify: \(\frac{x^2+3 x+2}{x^2+5 x+6} \times \frac{x^2+2 x-3}{x^2-4}\)
Solution:
Given \(\frac{x^2+3 x+2}{x^2+5 x+6} \times \frac{x^2+2 x-3}{x^2-4}\).
= \(\frac{x^2+3 x+2}{x^2+5 x+6} \times \frac{x^2+2 x-3}{x^2-4}\)
= \(\frac{x^2+2 x+x+2}{x^2+3 x+2 x+6} \times \frac{x^2+3 x-x-3}{x^2-4}\)
= \(\frac{x(x+2)+1(x+2)}{x(x+3)+2(x+3)} \times \frac{x(x+3)-1(x+3)}{(x+2)(x-2)}\)
= \(\frac{(x+1)(x+2)}{(x+2)(x+3)} \times \frac{(x+3)(x-1)}{(x+2)(x-2)}\)
= \(\frac{(x+1)(x-1)}{(x+2)(x-2)}=\frac{x^2-1}{x^2-4}\)
Example: 9
Simplify: \(\frac{x-2 y}{x y}+\frac{3 y-a}{a y}+\frac{3 x-2 a}{a x} \text {. }\)
Solution:
Given \(\frac{x-2 y}{x y}+\frac{3 y-a}{a y}+\frac{3 x-2 a}{a x} \text {. }\).
= \(\frac{x-2 y}{x y}+\frac{3 y-a}{a y}+\frac{3 x-2 a}{a x}\)
= \(\frac{a(x-2 y)+x(3 y-a)-y(3 x-2 a)}{a x y}\)
= \(\frac{a x-2 a y+3 x y-a x-3 x y+2 a y}{a x y}\)
= \(\frac{0}{a x y}=0\)
Example 10
Simplify: \(\frac{1}{x^2-3 x+2}+\frac{1}{x^2-5 x+6}+\frac{1}{x^2-4 x+3}\)
Solution:
Given
\(\frac{1}{x^2-3 x+2}+\frac{1}{x^2-5 x+6}+\frac{1}{x^2-4 x+3}\).
x2 – 3x + 2
= x2 – 2x – x +2
= x (x – 2) – 1 (x – 2)
= (x – 2)(x – 1)
x2 – 5x + 6 = x2 – 3x – 2x + 6
= x ( x – 3) – 2 (x – 3)
= (x – 3)(x – 2)
x2 – 4x + 3 = x2 -3x – x + 3
= x (x – 3) – 1 (x – 3)
= (x – 3)(x – 1)
∴ the given expression
= \(\frac{1}{(x-1)(x-2)}+\frac{1}{(x-2)(x-3)}+\frac{1}{(x-3)(x-1)}\)
= \(\frac{x-3+x-1+x-2}{(x-1)(x-2)(x-3)}\)
= \(\frac{3 x-6}{(x-1)(x-2)(x-3)}=\frac{3(x-2)}{(x-1)(x-2)(x-3)}\)
= \(\frac{3}{(x-1)(x-3)}=\frac{3}{x^2-4 x+3}\)
Conceptual Questions on Fraction Operations
Example 11
Simplify: \(\frac{\frac{x^2}{5-x}+\frac{y^2}{5-y}+\frac{z^2}{5-z}+x+y+z}{\frac{x}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}}\)
Solution:
Given
\(\frac{\frac{x^2}{5-x}+\frac{y^2}{5-y}+\frac{z^2}{5-z}+x+y+z}{\frac{x}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}}\).
= \(\frac{\frac{x^2}{5-x}+x+\frac{y^2}{5-y}+y+\frac{z^2}{5-z}+z}{\frac{x}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}}\)
= \(\frac{\frac{x^2+5 x-x^2}{5-x}+\frac{y^2+5 y-y^2}{5-y}+\frac{z^2+5 z-z^2}{5-z}}{\frac{x}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}}\)
= \(\frac{\frac{5 x}{5-x}+\frac{5 y}{5-y}+\frac{5 z}{5-z}}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}\)
= \(\text { 5. } \frac{\frac{x}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}}{\frac{x}{5-x}+\frac{y}{5-y}+\frac{z}{5-z}}\)
= 5
Example 12
Simplify: \(\left(a+\frac{a x}{a-x}\right) \times\left(a-\frac{a x}{a+x}\right) \times \frac{a^2-x^2}{a^2+x^2}\)
Solution:
The given Expression \(\left(a+\frac{a x}{a-x}\right) \times\left(a-\frac{a x}{a+x}\right) \times \frac{a^2-x^2}{a^2+x^2}\)
= \(\left(a+\frac{a x}{a-x}\right) \times\left(a-\frac{a x}{a+x}\right) \times \frac{a^2-x^2}{a^2+x^2}\)
= \(\frac{a^2-a x+a x}{a-x} \times \frac{a^2+a x-a x}{a+x} \times \frac{a^2-x^2}{a^2+x^2}\)
= \(\frac{a^2}{a-x} \times \frac{a^2}{a+x} \times \frac{a^2-x^2}{a^2+x^2}\)
= \(\frac{a^4 \times\left(a^2-x^2\right)}{\left(a^2-x^2\right)\left(a^2+x^2\right)}=\frac{a^4}{a^2+x^2}\)
Practice Problems on Simplifying Fractions
Example 13
Simplify: \(\frac{1+8 x^3}{(2-x)^2} \times \frac{4 x-x^3}{1-4 x^2} \div \frac{(1-2 x)^2+2 x}{2-5 x+2 x^2}\)
Solution:
The given expression \(\frac{1+8 x^3}{(2-x)^2} \times \frac{4 x-x^3}{1-4 x^2} \div \frac{(1-2 x)^2+2 x}{2-5 x+2 x^2}\)
= \frac{1+8 x^3}{(2-x)^2} \times \frac{4 x-x^3}{1-4 x^2} \times \frac{2-5 x+2 x^2}{(1-2 x)^2+2 x}
= \frac{1+(2 x)^3}{(2-x)^2} \times \frac{x\left(4-x^2\right)}{1-(2 x)^2} \times \frac{2-4 x-x+2 x^2}{1-4 x+4 x^2+2 x}
= \frac{(1+2 x)\left(1-2 x+4 x^2\right)}{(2-x)^2} \times \frac{x(2+x)(2-x)}{(1+2 x)(1-2 x)}\times \frac{2(1-2 x)-x(1-2 x)}{1-2 x+4 x^2}
= \frac{(1+2 x)\left(1-2 x+4 x^2\right)}{(2-x)^2} \times \frac{x(2+x)(2-x)}{(1+2 x)(1-2 x)}
= x(2 + x)
Example 14
Simplify: \(\frac{(b-c)^2}{(c-a)(a-b)}+\frac{(c-a)^2}{(a-b)(b-c)}+\frac{(a-b)^2}{(b-c)(c-a)} .\)
Solution:
The given expression
\(\frac{(b-c)^2}{(c-a)(a-b)}+\frac{(c-a)^2}{(a-b)(b-c)}+\frac{(a-b)^2}{(b-c)(c-a)} .\)= \(\frac{(b-c)^3+(c-a)^3+(a-b)^3}{(a-b)(b-c)(c-a)}\)
= \(\frac{b^3-c^3-3 b c(b-c)+c^3-a^3-3 c a(c-a)+a^3-b^3-3 a b(a-b)}{(a-b)(b-c)(c-a)}\)
= \(-\frac{3}{(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})}\{\mathrm{ab}(\mathrm{a}-\mathrm{b})+\mathrm{bc}(\mathrm{b}-\mathrm{c})+\mathrm{ca}(\mathrm{c}-\mathrm{a})\}\)
= \(-\frac{3}{(a-b)(b-c)(c-a)}\left\{a b(a-b)+b^2 c-b c^2+c^2 a-a^2 c\right\}\)
= \(-\frac{3}{(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})}\left\{\mathrm{ab}(\mathrm{a}-\mathrm{b})-\mathrm{c}\left(\mathrm{a}^2-\mathrm{b}^2\right)+\mathrm{c}^2(\mathrm{a}-\mathrm{b})\right\}\)
= \(-\frac{3}{(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})}\left\{\mathrm{ab}(\mathrm{a}-\mathrm{b})-\mathrm{c}(\mathrm{a}+\mathrm{b})(\mathrm{a}-\mathrm{b})+\mathrm{c}^2(\mathrm{a}-\mathrm{b})\right\}\)
= \(-\frac{3(\mathrm{a}-\mathrm{b})}{(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})}\left\{\mathrm{ab}-\mathrm{ac}-\mathrm{bc}+\mathrm{c}^2\right\}\)
= \(-\frac{3(\mathrm{a}-\mathrm{b})}{(\mathrm{a}-\mathrm{b})(\mathrm{b}-\mathrm{c})(\mathrm{c}-\mathrm{a})}\{\mathrm{a}(\mathrm{b}-\mathrm{c})-\mathrm{c}(\mathrm{b}-\mathrm{c})\}\)
= \(-\frac{3(a-b)(b-c)(a-c)}{(a-b)(b-c)(c-a)}=\frac{3(a-b)(b-c)(c-a)}{(a-b)(b-c)(c-a)}=3\)
Example 15
Simplify: \(\frac{1}{x-1}+\frac{1}{x+1}+\frac{2 x}{x^2+1}+\frac{4 x^3}{x^4+1}\)
Solution:
Given
\(\frac{1}{x-1}+\frac{1}{x+1}+\frac{2 x}{x^2+1}+\frac{4 x^3}{x^4+1}\).
\(\frac{1}{x-1}+\frac{1}{x+1}+\frac{2 x}{x^2+1}+\frac{4 x^3}{x^4+1}=\frac{x+1+x-1}{(x-1)(x+1)}+\frac{2 x}{x^2+1}+\frac{4 x^3}{x^4+1}\)= \(\frac{2 x}{x^2-1}+\frac{2 x}{x^2+1}+\frac{4 x^3}{x^4+1}=\frac{2 x^3+2 x+2 x^3-2 x}{\left(x^2-1\right)\left(x^2+1\right)}+\frac{4 x^3}{x^4+1}\)
= \(\frac{4 x^3}{x^4-1}+\frac{4 x^3}{x^4+1}=\frac{4 x^7+4 x^3+4 x^7-4 x^3}{\left(x^4-1\right)\left(x^4+1\right)}\)
= \(\frac{8 x^7}{\left(x^4\right)^2-(1)^2}=\frac{8 x^7}{x^8-1}\)
Examples of Simplifying Mixed Numbers
Example 16
Simplify: \(\frac{(b+c)\left(b^2+c^2-a^2\right)}{2 b c}+\frac{(c+a)\left(c^2+a^2-b^2\right)}{2 c a}+\frac{(a+b)\left(a^2+b^2-c^2\right)}{2 a b}\)
Solution:
Given
\(\frac{(b+c)\left(b^2+c^2-a^2\right)}{2 b c}+\frac{(c+a)\left(c^2+a^2-b^2\right)}{2 c a}+\frac{(a+b)\left(a^2+b^2-c^2\right)}{2 a b}\)= \(\frac{1}{2}\left(\frac{\mathrm{b}+\mathrm{c}}{\mathrm{bc}}\right)\left(\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2\right)+\frac{1}{2}\left(\frac{\mathrm{c}+\mathrm{a}}{\mathrm{ca}}\right)\left(\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\right)+\frac{1}{2}\left(\frac{\mathrm{a}+\mathrm{b}}{\mathrm{ab}}\right)\left(\mathrm{a}^2+\mathrm{b}^2-\mathrm{c}^2\right)\)
= \(\frac{1}{2}\left(\frac{1}{\mathrm{c}}+\frac{1}{\mathrm{~b}}\right)\left(\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2\right)+\frac{1}{2}\left(\frac{1}{\mathrm{a}}+\frac{1}{\mathrm{c}}\right)\left(\mathrm{c}^2+\mathrm{a}^2-\mathrm{b}^2\right)+\frac{1}{2}\left(\frac{1}{\mathrm{~b}}+\frac{1}{\mathrm{a}}\right)\left(\mathrm{a}^2+\mathrm{b}^2-\mathrm{c}^2\right)\)
= \(\frac{1}{2}\left[\begin{array}{c}
\frac{1}{c}\left(b^2+c^2-a^2+c^2+a^2-b^2\right)+\frac{1}{b}\left(b^2+c^2-a^2+a^2+b^2-c^2\right) \\
+\frac{1}{a}\left(c^2+a^2-b^2+a^2+b^2-c^2\right)
\end{array}\right]\)
= \(\frac{1}{2}\left[\frac{1}{c} 2 \mathrm{c}^2+\frac{1}{\mathrm{~b}} 2 \mathrm{~b}^2+\frac{1}{\mathrm{a}} 2 \mathrm{a}^2\right]=\frac{1}{2}[2 \mathrm{c}+2 \mathrm{~b}+2 \mathrm{a}]=\frac{2}{2}[\mathrm{c}+\mathrm{b}+\mathrm{a}]=\mathrm{a}+\mathrm{b}+\mathrm{c}\)