Surds and Indices Formulas

Surds and Indices Rule

Surds and Indices Formulas

• (a + b)(a – b) = (a² – b²)
• (a + b)² = (a² + b² + 2ab)
• (a – b)² = (a² + b²- 2ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ + b³) = (a + b)(a² – ab + b²)
• (a³ – b³) = (a – b)(a²+ ab + b²)
• (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc.

Questions and answer based on Formulas 

Question 1 :

Find the value of (3x + 2y)^{2} using (a + b)^{2} formula.

Solution:

To find: The value of (3x + 2y)^{2}
Let us assume that a = 3x and b = 2y.
We will substitute these values in (a + b)^{2} formula:
(a + b)^{2} = a^{2} + 2ab + b^{2}
(3x + 2y)^{2} =(3x)^{2} + 2(3x)(2y) + (2y)^2
9x^{2} + 12xy + 4y^{2}

Answer:
(3x + 2y)^{2} = 9x^{2} + 12xy + 4y^{2}

Question 2 :

Solve the following expression using suitable algebraic identity: (2x + 3y)^{3}

Solution:

To find:
(2x + 3y)^{3}
Using (a + b)^{3}Formula,
(a + b)^{3} = a^{3} + 3a2b + 3ab2 + b^{3}
(2x)^{3} + 3 × (2x)2 × 3y + 3 × (2x) × (3y)2 + (3y)^{3}
8x^{3} + 36x2y + 54xy2 + 27y^{3}

Answer:
(2x + 3y)^{3} = 8x^{3} + 36x2y + 54xy2 + 27y^{3}

Question 3:

Rishi kapoor wants to know that by how much does \sqrt{12} + \sqrt{18} exceed \sqrt{3} + \sqrt{2} ?

Solution : 

(\sqrt{12} + \sqrt{18}) – (\sqrt{3} + \sqrt{2} )
(2\sqrt{3} – \sqrt{3}) – (3\sqrt{2} – \sqrt{2} )
\sqrt{3} + 2\sqrt{2}

Question 4 :

Ranbeer kapoor wants to know the value of (256)^{0.16} \times (16)^{0.18} :

Solution :

Expression = (256)^{0.16} \times (16)^{0.18}
= (4)^{4}\times 0.16 \times (4)^{2}\times 0.18
= (4)^{0.64}\times (4)^{0.36}
= (4)^{0.64+0.36}
= (4)^{1} = 4

Question 5:

Sunil shetty wants to know that (0.04)^{–(1.5)} is equal to

Solution :

Expression = (0.04)^{–1.5}
= \frac{1}{ 0.04^{1.5}}

= \frac{1}{0.04^{\frac{3}{2}}} 
= \frac{1}{\sqrt{0. 0000064}}
= \frac{1}{0.008} =\frac{1000}{8}= 125