Maharashtra Board Class 8 Maths part 1 Chapter 3 Indices and Cube root PDF Download

Indices And Cube Root

Let's Recall

In earlier standards, we have learnt about Indices and laws of indices.

The product 2 × 2 × 2 × 2 × 2, can be expressed as 2⁵, in which 2 is the base, 5 is the index and 2⁵ is the index form of the number.

Laws of indices: If m and n are integers, then

(i) a^m × aⁿ = a^m+n

(ii) a^m ÷ aⁿ = a^m-n

(iii) (a × b)^m = a^m × b^m

(iv) a⁰ = 1

(v) a^-m = \(\frac{1}{a^m}\)

(vi) (a^m)ⁿ = a^mn

(vii) \(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\)

(viii) \(\left(\frac{a}{b}\right)^{-m} = \left(\frac{b}{a}\right)^m\)

Using laws of indices, write proper numbers in the following boxes.

(i) 3⁵ × 3² = 3 ____

(ii) 3⁷ ÷ 3⁹ = 3 ____

(iii) (3⁴)⁵ = 3 ____

(iv) 5^-3 = \(\frac{1}{5}\) ____

(v) 5⁰ = ____

(vi) 5¹ = ____

(vii) (5 × 7)² = 5 ____ × 7 ____

(viii) \(\left(\frac{5}{7}\right)^3 = \frac{____^3}{____^3}\)

(ix) \(\left(\frac{5}{7}\right)^{-3} = \left(\frac{____}{____}\right)^3\)

Teacher's Note

Indices help us write big numbers in a short way. For example, 10 million rupees can be written as 10⁷ in a simple form.

Exam Trick

Remember: When you multiply same numbers with indices, add the powers. When you divide, subtract the powers. This is like combining or separating groups of items.

Points to Remember

Index form helps us write very large or very small numbers in short.
When multiplying same bases, add the powers together.
When dividing same bases, subtract the powers.
Any number with power 0 always equals 1.
A negative power means the number goes in the denominator.

Let's Learn

Meaning Of Numbers With Rational Indices

(I) Meaning Of The Numbers When The Index Is A Rational Number Of The Form \(\frac{1}{n}\)

Let us see the meaning of indices in the form of rational numbers such as \(\frac{1}{2}, \frac{1}{3}, \frac{1}{5}, ..., \frac{1}{n}\).

To show the square of a number, the index is written as 2 and to show the square root of a number, the index is written as \(\frac{1}{2}\).

For example, square root of 25, is written as \(\sqrt{25}\) using the radical sign '√'. Using index, it is expressed as \(25^{\frac{1}{2}}\). Therefore \(\sqrt{25} = 25^{\frac{1}{2}}\).

In general, square of a can be written as a² and square root of a is written as \(\sqrt{a}\) or a or \(a^{\frac{1}{2}}\).

Similarly, cube of a is written as a³ and cube root of a is written as \(\sqrt[3]{a}\) or \(a^{\frac{1}{3}}\).

For example, 4³ = 4 × 4 × 4 = 64. Therefore cube root of 64 can be written as \(\sqrt[3]{64}\) or \((64)^{\frac{1}{3}}\). Note that, \((64)^{\frac{1}{3}}\) = 4.

3 × 3 × 3 × 3 × 3 = 3⁵ = 243. That is 5th power of 3 is 243.

Conversely, 5th root of 243 is expressed as \((243)^{\frac{1}{5}}\) or \(\sqrt[5]{243}\). Hence, \((243)^{\frac{1}{5}}\) = 3.

In general nth root of a is expressed as \(a^{\frac{1}{n}}\).

For example, (i) \(128^{\frac{1}{7}}\) = 7th root of 128, (ii) \(900^{\frac{1}{12}}\) = 12th root of 900, etc.

Note that, If \(10^{\frac{1}{5}}\) = x then x⁵ = 10.

Practice Set 3.1

1. Express the following numbers in index form.

(1) Fifth root of 13

(2) Sixth root of 9

(3) Square root of 256

(4) Cube root of 17

(5) Eighth root of 100

(6) Seventh root of 30

2. Write in the form 'nth root of a' in each of the following numbers.

(1) \(81^{\frac{1}{4}}\)

(2) \(49^{\frac{1}{2}}\)

(3) \(15^{\frac{1}{5}}\)

(4) \(512^{\frac{1}{9}}\)

(5) \(100^{\frac{1}{19}}\)

(6) \(6^{\frac{1}{7}}\)

Teacher's Note

Rational indices are just another way to write roots of numbers. When you see \(a^{\frac{1}{n}}\), it simply means the nth root of a, just like \(\sqrt[n]{a}\).

Exam Trick

Remember: The bottom number in the fraction index is always the root. So \(a^{\frac{1}{3}}\) means cube root, \(a^{\frac{1}{5}}\) means 5th root. The denominator tells you which root to take.

Points to Remember

\(a^{\frac{1}{2}}\) means square root of a.
\(a^{\frac{1}{3}}\) means cube root of a.
\(a^{\frac{1}{n}}\) means nth root of a.
The denominator of the fraction index tells us which root to find.
Square root sign √ is the same as raising to power \(\frac{1}{2}\).

(II) The Meaning Of Numbers, Having Index In The Rational Form \(\frac{m}{n}\)

We know that 8² = 64.

Cube root of 64 is = \((64)^{\frac{1}{3}}\) = \((8^2)^{\frac{1}{3}}\) = 4

Therefore cube root of square of 8 is 4 .......... (I)

Similarly, cube root of 8 = \(8^{\frac{1}{3}}\) = 2

Therefore square of cube root of 8 is \(\left(8^{\frac{1}{3}}\right)^2\) = 2² = 4 ..........(II)

From (I) and (II)

cube root of square of 8 = square of cube root of 8. Using indices, \((8^2)^{\frac{1}{3}}\) = \(\left(8^{\frac{1}{3}}\right)^2\).

The rules for rational indices are the same as those for integral indices.

Therefore using the rule (a^m)ⁿ = a^mn, we get \((8^2)^{\frac{1}{3}}\) = \(\left(8^{\frac{1}{3}}\right)^2\) = \(8^{\frac{2}{3}}\).

From this we get two meanings of the number \(8^{\frac{2}{3}}\).

(i) \(8^{\frac{2}{3}}\) = \((8^2)^{\frac{1}{3}}\) i.e. cube root of square of 8.

(ii) \(8^{\frac{2}{3}}\) = \(\left(8^{\frac{1}{3}}\right)^2\) i.e. square of cube root of 8.

Similarly, \(27^{\frac{4}{5}}\) = \((27^4)^{\frac{1}{5}}\) means 'fifth root of fourth power of 27',

and \(27^{\frac{4}{5}}\) = \(\left(27^{\frac{1}{5}}\right)^4\) means 'fourth power of fifth root of 27'.

Generally we can express two meanings of the number \(a^{\frac{m}{n}}\).

\(a^{\frac{m}{n}}\) = \((a^m)^{\frac{1}{n}}\) means 'nth root of mth power of a'.

\(a^{\frac{m}{n}}\) = \(\left(a^{\frac{1}{n}}\right)^m\) means 'mth power of nth root of a'.

Practice Set 3.2

1. Complete the following table.

2. Write the following numbers in the form of rational indices.

(1) Square root of 5th power of 121.

(2) Cube of 4th root of 324

(3) 5th root of square of 264

(4) Cube of cube root of 3

Teacher's Note

Rational indices like \(\frac{2}{3}\) are useful because they let us work with both powers and roots at the same time. This is very helpful in solving difficult math problems.

Exam Trick

Remember: In \(a^{\frac{m}{n}}\), the top number m is the power and the bottom number n is the root. So \(8^{\frac{2}{3}}\) means: first take cube root (bottom number), then square it (top number).

Points to Remember

\(a^{\frac{m}{n}}\) can be written in two ways: as nth root of mth power, or as mth power of nth root.
The numerator tells us which power to use.
The denominator tells us which root to use.
Both ways of writing give the same answer.
This makes calculations easier and more flexible.

Let's Recall

4 × 4 = 16 implies 4² = 16, also (-4)² = 16 which indicates that the number 16 has two square roots; one positive and the other negative. Conventionally, positive root of 16 is shown as \(\sqrt{16}\) and negative root of 16 is shown as \(-\sqrt{16}\). Hence \(\sqrt{16}\) = 4 and \(-\sqrt{16}\) = -4.

Every positive number has two square roots.

Square root of zero is zero.

Teacher's Note

In India, when we buy land, we measure it in square meters. To find the side of a square plot, we use square root. This is how roots are used in real life.

Exam Trick

Remember: Every positive number has TWO square roots - one positive and one negative. But \(\sqrt{16}\) always means only the positive one, which is 4. The negative one is written as \(-\sqrt{16}\).

Points to Remember

Square of any number, positive or negative, is always positive.
Every positive number has two square roots.
The symbol √ shows