Algebraic Identities Worksheet 3
1. Find the following by using identity - III:
(i) \( (2x+7y)(2x-7y) \)
Solution
\[ \begin{align*} \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 2x \, , \, b = 7y \\ \\ & = (2x)^2 - (7y)^2 \\ & = \color{red} 4x^2 - 49y^2 \end{align*} \]
(ii) \( (5ab - 8c)(5ab + 8c) \)
Solution
\[ \begin{align*} \color{green} (a-b)(a+b) &= \color{green} a^2 - b^2 \\ & a = 5ab \, , \, b = 8c \\ \\ & = (5ab)^2 - (8c)^2 \\ & = \color{red} 25a^2b^2 - 64c^2 \end{align*} \]
(iii) \( (4p^2 + q^2)(4p^2 - q^2) \)
Solution
\[ \begin{align*} \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 4p^2 \, , \, b = q^2 \\ \\ & = (4p^2)^2 - (q^2)^2 \\ & = \color{red} 16p^4 - q^4 \end{align*} \]
(iv) \( \left( \frac{x}{3} - \frac{y}{2} \right)\left( \frac{x}{3} + \frac{y}{2} \right) \)
Solution
\[ \begin{align*} \color{green} (a-b)(a+b) &= \color{green} a^2 - b^2 \\ & a = \frac{x}{3} \, , \, b = \frac{y}{2} \\ \\ & = \left( \frac{x}{3} \right)^2 - \left( \frac{y}{2} \right)^2 \\ \\ & = \color{red} \frac{x^2}{9} - \frac{y^2}{4} \end{align*} \]
(v) \( (0.1m - 0.2n)(0.1m + 0.2n) \)
Solution
\[ \begin{align*} \color{green} (a-b)(a+b) &= \color{green} a^2 - b^2 \\ & a = 0.1m \, , \, b = 0.2n \\ \\ & = (0.1m)^2 - (0.2n)^2 \\ & = \color{red} 0.01m^2 - 0.04n^2 \end{align*} \]
(vi) \( (a^3 + b^3)(a^3 - b^3) \)
Solution
\[ \begin{align*} \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = a^3 \, , \, b = b^3 \\ \\ & = (a^3)^2 - (b^3)^2 \\ & = \color{red} a^6 - b^6 \end{align*} \]
2. Evaluate the following by using identity - III:
(i) \( 81^2 - 19^2 \)
Solution
\[ \begin{align*} \color{green} a^2 - b^2 &= \color{green} (a+b)(a-b) \\ & a = 81 \, , \, b = 19 \\ \\ & = (81+19)(81-19) \\ & = 100 \times 62 \\ & = \color{red} 6200 \end{align*} \]
(ii) \( 290^2 - 210^2 \)
Solution
\[ \begin{align*} \color{green} a^2 - b^2 &= \color{green} (a+b)(a-b) \\ & a = 290 \, , \, b = 210 \\ \\ & = (290+210)(290-210) \\ & = 500 \times 80 \\ & = \color{red} 40000 \end{align*} \]
(iii) \( 58^2 - 12^2 \)
Solution
\[ \begin{align*} \color{green} a^2 - b^2 &= \color{green} (a+b)(a-b) \\ & a = 58 \, , \, b = 12 \\ \\ & = (58+12)(58-12) \\ & = 70 \times 46 \\ & = \color{red} 3220 \end{align*} \]
(iv) \( 176^2 - 24^2 \)
Solution
\[ \begin{align*} \color{green} a^2 - b^2 &= \color{green} (a+b)(a-b) \\ & a = 176 \, , \, b = 24 \\ \\ & = (176+24)(176-24) \\ & = 200 \times 152 \\ & = \color{red} 30400 \end{align*} \]
(v) \( 367^2 - 33^2 \)
Solution
\[ \begin{align*} \color{green} a^2 - b^2 &= \color{green} (a+b)(a-b) \\ & a = 367 \, , \, b = 33 \\ \\ & = (367+33)(367-33) \\ & = 400 \times 334 \\ & = \color{red} 133600 \end{align*} \]
(vi) \( 545^2 - 445^2 \)
Solution
\[ \begin{align*} \color{green} a^2 - b^2 &= \color{green} (a+b)(a-b) \\ & a = 545 \, , \, b = 445 \\ \\ & = (545+445)(545-445) \\ & = 990 \times 100 \\ & = \color{red} 99000 \end{align*} \]
3. Simplify the following products by expressing these as difference of two squares:
(i) \( 107 \times 93 \)
Solution
\[ \begin{align*} 107 \times 93 & = (100+7)(100-7) \\ \\ \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 100 \, , \, b = 7 \\ \\ & = (100)^2 - (7)^2 \\ & = 10000 - 49 \\ & = \color{red} 9951 \end{align*} \]
(ii) \( 211 \times 189 \)
Solution
\[ \begin{align*} 211 \times 189 & = (200 + 11)(200 - 11) \\ \\ \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 200 \, , \, b = 11 \\ \\ & = (200)^2 - (11)^2 \\ & = 40000 - 121 \\ & = \color{red} 39879 \end{align*} \]
(iii) \( 195 \times 205 \)
Solution
\[ \begin{align*} 195 \times 205 & = (200 - 5)(200 + 5) \\ \\ \color{green} (a-b)(a+b) &= \color{green} a^2 - b^2 \\ & a = 200 \, , \, b = 5 \\ \\ & = (200)^2 - (5)^2 \\ & = 40000 - 25 \\ & = \color{red} 39975 \end{align*} \]
(iv) \( 308 \times 292 \)
Solution
\[ \begin{align*} 308 \times 292 & = (300 + 8)(300 - 8) \\ \\ \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 300 \, , \, b = 8 \\ \\ & = (300)^2 - (8)^2 \\ & = 90000 - 64 \\ & = \color{red} 89936 \end{align*} \]
(v) \( 12.4 \times 11.6 \)
Solution
\[ \begin{align*} 12.4 \times 11.6 & = (12 + 0.4)(12 - 0.4) \\ \\ \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 12 \, , \, b = 0.4 \\ \\ & = (12)^2 - (0.4)^2 \\ & = 144 - 0.16 \\ & = \color{red} 143.84 \end{align*} \]
(vi) \( 30.9 \times 29.1 \)
Solution
\[ \begin{align*} 30.9 \times 29.1 & = (30 + 0.9)(30 - 0.9) \\ \\ \color{green} (a+b)(a-b) &= \color{green} a^2 - b^2 \\ & a = 30 \, , \, b = 0.9 \\ \\ & = (30)^2 - (0.9)^2 \\ & = 900 - 0.81 \\ & = \color{red} 899.19 \end{align*} \]