Educational

Evaluate Using Identity Evaluate Using Identity \[ 991 \times 1009 \] Solution: Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ 991 \times 1009 = (1000-9)(1000+9) \] \[ = (1000)^2 – (9)^2 \] \[ = 1000000 – 81 \] \[ = 999919 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate Using Identity \[ (0.98)^2 \] Solution: Using identity: \[ (a-b)^2 = a^2 – 2ab + b^2 \] \[ (0.98)^2 = (1-0.02)^2 \] \[ = (1)^2 – 2(1)(0.02) + (0.02)^2 \] \[ = 1 – 0.04 + 0.0004 \] \[ = 0.9604 \] Next Question / Full Exercise

Evaluate Using Identity Evaluate Using Identity \[ (399)^2 \] Solution: Using identity: \[ (a-b)^2 = a^2 – 2ab + b^2 \] \[ 399^2 = (400-1)^2 \] \[ = (400)^2 – 2(400)(1) + (1)^2 \] \[ = 160000 – 800 + 1 \] \[ = 159201 \] Next Question / Full Exercise

Evaluate Expression Evaluate \[ (1.5x^2 – 0.3y^2)(1.5x^2 + 0.3y^2) \] Solution: Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ (1.5x^2 – 0.3y^2)(1.5x^2 + 0.3y^2) \] \[ = (1.5x^2)^2 – (0.3y^2)^2 \] \[ = 2.25x^4 – 0.09y^4 \] Next Question / Full Exercise

Evaluate Expression Evaluate \[ (a – 0.1)(a + 0.1) \] Solution: Using identity: \[ (a-b)(a+b)=a^2-b^2 \] \[ (a – 0.1)(a + 0.1) \] \[ = a^2 – (0.1)^2 \] \[ = a^2 – 0.01 \] Next Question / Full Exercise

Evaluate Expression Evaluate \[ (a^2b – b^2a)^2 \] Solution: Using identity: \[ (a-b)^2 = a^2 – 2ab + b^2 \] \[ (a^2b – b^2a)^2 \] \[ = (a^2b)^2 – 2(a^2b)(b^2a) + (b^2a)^2 \] \[ = a^4b^2 – 2a^3b^3 + a^2b^4 \] Next Question / Full Exercise

Evaluate Expression Evaluate \[ (2x + y)(2x – y) \] Solution: Using identity: \[ (a+b)(a-b)=a^2-b^2 \] \[ (2x + y)(2x – y) \] \[ = (2x)^2 – y^2 \] \[ = 4x^2 – y^2 \] Next Question / Full Exercise

Evaluate Expression Evaluate \[ \left(2x – \frac{1}{x}\right)^2 \] Solution: Using identity: \[ (a-b)^2 = a^2 – 2ab + b^2 \] \[ \left(2x – \frac{1}{x}\right)^2 = (2x)^2 – 2\left(2x\right)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 \] \[ = 4x^2 – 4 + \frac{1}{x^2} \] Next Question / Full Exercise

Find the Value of a Find the value of \(a\) If \[ \sqrt{13 – a\sqrt{10}} = \sqrt{8} + \sqrt{5} \] (a) −5 \quad (b) −6 \quad (c) −4 \quad (d) −2 Solution: \[ \sqrt{8} = 2\sqrt{2} \Rightarrow \sqrt{8} + \sqrt{5} = 2\sqrt{2} + \sqrt{5} \] \[ (2\sqrt{2} + \sqrt{5})^2 = 8 + 5 + 4\sqrt{10}

Find the Value Find the value If \[ x = \sqrt{6} + \sqrt{5} \] (a) \(2\sqrt{6}\) \quad (b) \(2\sqrt{5}\) \quad (c) 24 \quad (d) 20 Solution: \[ \frac{1}{x} = \sqrt{6} – \sqrt{5} \] \[ x + \frac{1}{x} = (\sqrt{6} + \sqrt{5}) + (\sqrt{6} – \sqrt{5}) = 2\sqrt{6} \] \[ x^2 + \frac{1}{x^2} – 2 =