Maharashtra Board Class 7 Maths part 1 Chapter 6 Indices PDF Download

Indices

Let's Recall

Each of 7 children was given 4 books.

Total notebooks = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 28 notebooks

Here, addition is the operation that is carried out repeatedly.

Addition of the same number again and again can be shown as a multiplication.

Total notebooks = 4 + 4 + 4 + 4 + 4 + 4 + 4 = 4 × 7 = 28

Base and Index

Let us see how the multiplication of a number by itself several times is expressed in short.

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2: Here, 2 is multiplied by itself 8 times.

This is written as \(2^8\) in short. This is the index form of the multiplication.

Here, 2 is called the base and 8, the index or the exponent.

Example 5 × 5 × 5 × 5 = \(5^4\) Here \(5^4\) is in the index form.

In the number \(5^4\), 5 is the base and 4 is the index.

This is read as '5 raised to the power 4' or '5 raised to 4', or 'the 4th power of 5'.

Generally, if a is any number, a × a × a ×.......... (m times) = \(a^m\)

Read \(a^m\) as 'a raised to the power m' or 'the m^th power of a'.

Here m is a natural number.

\(5^4 = 5 \times 5 \times 5 \times 5 = 625\). Or, the value of the number \(5^4 = 625\).

Similarly, \(\left(\frac{2}{3}\right)^3 = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} = \frac{8}{27}\) means that the value of \(\left(\frac{2}{3}\right)^3\) is \(\frac{8}{27}\).

Note that \(7^1 = 7\), \(10^1 = 10\). The first power of any number is that number itself. If the power or index of a number is 1, the convention is not to write it.

Thus \(5^1 = 5\), \(a^1 = a\).

Teacher's Note

In our India Post Office, we use indices to show postal codes and bank account numbers. When we write numbers in a short form using powers, we save time and space, just like how we use short codes for addresses.

Exam Trick

Remember: The base is the number at the bottom, and the index is the small number at the top. Think of it like a building - the base is the ground floor, and the index tells you how many times to go up!

Points to Remember

\(a^m\) means a is multiplied by itself m times.
The base is the number being multiplied.
The index (or exponent) tells how many times to multiply.
\(a^1 = a\) always - we do not write the power 1.
\(a^m\) is read as 'a raised to the power m'.

Square and Cube

\(3^2 = 3 \times 3 = 9\) and \(5^3 = 5 \times 5 \times 5 = 125\)

\(3^2\) is read as '3 raised to 2' or '3 squared' or 'the square of 3'

\(5^3\) is read as '5 raised to 3' or '5 cubed' or 'the cube of 5'

Remember:

The second power of any number is the square of that number.

The third power of any number is the cube of that number.

Multiplication of Indices with the Same Base

Example \(2^4 \times 2^3\)

\(= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2\)

\(= 2^7\)

Therefore, \(2^4 \times 2^3 = 2^{4 + 3} = 2^7\)

Example \((-3)^2 \times (-3)^3\)

\(= (-3) \times (-3) \times (-3) \times (-3) \times (-3)\)

\(= (-3)^5\)

Therefore, \((-3)^2 \times (-3)^3 = (-3)^{2 + 3} = (-3)^5\)

Example \(\left(\frac{2}{5}\right)^2 \times \left(\frac{2}{5}\right)^3 = \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) \times \left(\frac{2}{5}\right) = \left(\frac{2}{5}\right)^5\)

Therefore, \(\left(\frac{2}{5}\right)^2 \times \left(\frac{2}{5}\right)^3 = \left(\frac{2}{5}\right)^{2 + 3} = \left(\frac{2}{5}\right)^5\)

Teacher's Note

When you multiply powers with the same base in our shops, we add the powers. For example, if you have 2 baskets with 3 apples each, and 3 baskets with 3 apples each, you can write it as \(3^2 \times 3^3 = 3^5\).

Exam Trick

Remember: When multiplying same bases, ADD the powers. Do not multiply them! Think: 2^3 × 2^2 = 2^(3+2) = 2^5, not 2^6.

Points to Remember

When bases are the same, add the powers when multiplying.
\(a^m \times a^n = a^{m+n}\)
This rule works for positive and negative numbers.
This rule works for fractions too.
The base must be the same to use this rule.

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