Surds And Indices

Surds and indices are commonly used mathematical terms. Surds represent numbers that can’t be simplified into whole numbers or integers. Indices are numbers raised as power. Indices or index numbers tell us how many times the base number is to be multiplied.

To simplify problems containing surds and indices, you require a good grasp of the concepts and the formulas associated with these terms. This article will discuss surds and indices, their definition, concepts, formulas, and questions.

Surds Definition

Surds are square roots of numbers that can’t be further simplified. These are roots of whole numbers that have an irrational value. If we try and simplify surds, we will get decimal values. Also, fractions are not a good way to represent surds.

Exampleof surds:√ 2 ,√3 ^{3√11, etc.}

Types of Surds

The  various categories of surds are as follows:

1. Simple Surds: Surds with a single term are termed as monomial or simple surds.

For example-√ 2 , ²√2

2. Pure Surds: Surds which have no rational factor other than unity are called complete or pure surds.

For example: √3, √7

3. Similar Surds: Surds with a common or similar surd factor are termed as similar or like surd.

For example: 3-√33

4. Mixed Surds: Surds which have a rational coefficient other than unity are mixed surds.

For example: 2√5, 3√6

5. Compound Surds: The algebraic addition or subtraction of a rational number and a simple surd.

For example: √3+√7

6. Binomial Surds: It represents a sum of two roots of rational numbers, atleast one of which is an irrational number.

For example: 4√3+√2 and 4√3-√2

Indices Definition

Indices or Index numbers are numbers raised as power. The power of a number represents the number of times the base has to be multiplied. Indices are the plural form of an index. Representation of an index number is as follows:

4²

Here, 4 is the base and 2 is the index. To simplify this:

4×4=16

Therefore, 4²=16

Indices can also be negative, fraction or 0. Indices can also be raised to a power, which we term as the power of a power.

Example of Negative indices:

3^-4=1/3⁴

Example of Fractional Indices:

5^ab=(b√5 )^a

Surds And Indices Formula

Here we will discuss the surds and indices formula used in the mathematical calculation of the same.

Formulas For Indices

Several laws govern the mathematical calculation of index numbers.

1. Law of Multiplication: When two terms with similar bases are multiplied, indices are added.

x^a×x^b=x^a+b

2. Law of Divison: When terms with similar bases are divided, indices are subtracted.

a^b÷a^c=a^b-c

3. Brackets: If there is a power bracket outside the bracket, the two powers are multiplied.

(a^m)ⁿ=a^mn

4. Power of 0: If a term has the power 0, its value becomes 1.

a⁰=1

5. Negative Indices: If the power of a term is negative, we write the reciprocal of the base, and the power becomes positive.

a^-b=1/a^b

6. Fractional Indices: If the index is fractional, the denominator becomes the root of the number, and the numerator becomes its power.

a^2⁄3=(3√a)²

7. Multiplication of Identical Indices: If two terms with identical indices are multiplied, their bases are multiplied.

a^b×c^b=(ac)^b

8. Divison of Identical indices: If two terms with identical indices are divided, their bases are divided.

a^b÷c^b=(a/c)^b

Formulas For Surds

1. Multiplication Rule

√a×b=√a×√b

2. Divison Rule

√a÷b=√a÷√b

3. Denominator Rule

a ÷√b=a ÷√b×√b ÷√b=a√b÷b

4. Addition Rule

a√c+b√c=(a+b)√c

5. Subtraction Rule

a√c-b√c=(a-b)√c

6. Rationalization Rule

c ⁄ a+b√n=c ⁄ a+b√n×a-b√n ⁄ a-bn

c ⁄ a-b√n=c ⁄ a-b√n×a+b√n ⁄ a+b√n

Surds And Indices Questions

Let us discuss some surds and indices questions to understand the concepts well.

Example 1: Rationalize the following term:

1 ⁄ (3√11)-(2√5)

Solution:

=1 ⁄ (3√11)-(2√5)× (3√11)+(2√5) ⁄ (3√11)+(2√5)

=(3√11)+(2√5)(3√11)²-(2√5)²

=(3√11)+(2√5) ⁄ 99-20

=(3√11)+(2√5) ⁄ 79

Example 2: Simplify (1331)^-2/3

Solution:

=(11³)^-2/3

=11^-2

=1/11²

=1/121

Conclusion

Surds and indices are important mathematical terms that are commonly used. They are used in other branches of mathematics as well. It is therefore essential to understand their concepts and formulas. Practice as many surds and indices questions as possible to solve them efficiently.