Surds and Indices Formulas

Formulas For Surds and Indices

Go through the entire page to know important formulas for Surds and Indices to solve the questions of Surds and Indices quickly.

Definition and Types of Surds and Indices

Definition of Surds and Indices

• Surds: Numbers which can be expressed in the form √p + √q , where p and q are natural numbers and not perfect squares.
• Irrational numbers which contain the radical sign (n√) are called as surds Hence, the numbers in the form of √3, 3√2, ……. n√x in other words
• For example : $\sqrt{3},$ it can’t be simplified.
$\sqrt{4},$ it can be simplified so it is not a surds.
• Indices: Indices refers to the power to which a number is raised. For example; 3²
• Surds and Indices formulas pages is very useful for solving the ques.. Prepinsta provide Surds and Indices Formulas and ques.

Types of Surds and Definitions

• Pure Surds:- Those surds which do not have factors other than 1. For example 2√3, 3√7
• Mixed Surds:- Those surds which do not have a factor of 1. For example √27 = 3√3, √50 = 5√2
• Similar Surds:- When the radicands of two surds are the same. For example 5√2 and 7√2
• Unlike Surds:- When the radicands are different. For example √2 and 2√5

Surds and Indices Rule

Rule NameSurds RuleIndices Rule
Multiplication Rulean * bn = (a*b)nan * am = a(m+n)
Division Rulean/ bn = (a/b)nam / an = a(m-n)
Power Rule

(an)m = (a)nm
n√a = a(1/n)

a(nm)) = anm
a-n = 1/(an)

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:
$(a + b)^{2} = a^{2} + 2ab + b^{2}$
$(3x + 2y)^{2} =(3x)^{2} + 2(3x)(2y) + (2y)^2$
$9x^{2} + 12xy + 4y^{2}$

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

Question 2 :

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

Solution:

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

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

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

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

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