A surd is a mathematical expression that includes the root function implied for any numeric value, be it a square root, a cube root, or any other root. Surds are helpful when we have to write irrational numbers without simplifying them into any format that cannot be understood easily and precisely. 

The primary reason behind the usage of surds is that the decimals that are generated after the simplification of irrational numbers follow recursion or do not terminate, which makes it difficult to write them precisely in decimal form.

For example, let us take √3. The value of √3 = 1.73205081. This value repeats  itself after the decimal place, so it is a good choice to leave it with the square root only and not complicate the calculations.

Any number or algebraic expression that is in the power or exponent place with respect to the base number is referred to as an index number. The index number is helpful in determining the exact number of iterations by which the base number is being multiplied by itself. 

Types Of Surds

• Simple Surds: – Simple Surds got their name from their characteristics because they have a single number, or we can call it monomial. 

For example, √2.

• Pure Surds: – Pure Surds are surds that do not have any rational factors, that is they are completely irrational. 

For example, √3 and √11.

• Similar Surds: – Similar surds are called such because these surds have a common root factor. 

For example, 3√2 and 7√2. 

• Mixed Surds: – Mixed Surds are surds that have a real number coefficient outside the root which makes them partially rational, and hence, they can be defined as a product of an irrational number with a rational number.

For example, 3√7 and 2√6.

• Compound Surds: – A compound surd is a surd that is the algebraic addition or subtraction of two or more than two surds.

For example, (√6 + √3)

Surds And Indices Formula

• √(a x b) = √a x √b

To simplify √18

18 = 2 x 9 = 2 x 32, since 9 is the perfect square inside the square root, so it can be broken down into 32.

Therefore, we can write √18 = √(32 x 2)

= √32 x √2

= 3 √2

• √(a / b) = √a / √b

For example:

√(12 / 49) = √12 / √49

=√(22 x 3) / 7

=√22 x √3 / 7

= 2√3 / 7

• a / √b = (a / √b) * ( √b  / √b ) = (a/b * √b)

This rule is called “rationalisation of the root.” We can remove the root from the denominator by multiplying it by the numerator and the denominator. 

For example:

Rationalising 5/√8 can be done by multiplying the numerator and denominator by √8

5/√8 = (5/√8) x (√8/√8) = 5√8/8

• a√b + c√b = (a + c)√b

For example: Simplifying 2√3 + 12√3 = (2 + 12)√3 = 14√3 

Surds & Indices Questions

• Write down the conjugate of 4√3 + √2?

4√3 – √2 is the conjugate of this expression. 

• Multiply √8 x √5.

√8 x √5 = √(8 x 5) = √40

• Divide √40 by √12.

(√40/√12) = (√10/√3) = (√10/√3 * √3/√3) = √30 / 3.

• 5³x-² =15625 , find x.

15625 can be written in exponential form as 56.

By comparing the exponential part of both LHS and RHS,

3x – 2 = 6.

3x = 8

x = 8/3.

• (13)4 x (13)x = (13)11, find x?

(13)4+x = (13)11

Therefore, 4 + x = 11.

x = 11 – 4 = 7.

Therefore, the value of x is 7. 

Conclusion

Surds and Indices are frequently asked in exams. The notion is related to algebra and is based on basic mathematics. The questions are basically asked around determining the power’s value. 

All you need is all these above-stated rules and formulas to solve every question that involves surds and indices. One of the most sought-out and easiest methods is rationalising the denominators to get rid of the root of the denominator. Sometimes, this method does not work the best. In those times, you should check whether any perfect squares can be extracted from the root.