This GSEB Class 7 Maths Notes Chapter 13 Exponents and Powers covers all the important topics and concepts as mentioned in the chapter.
Exponents and Powers Class 7 GSEB Notes
We come across numbers that are very large or very small in many situations. For example, the age of the universe in years, the mass of the earth in tons, the distance of the sun from the earth (in km), the size of bacteria etc. are numbers that are either very large or very small due to complexity of their measurement. Such numbers are generally approximate.
Therefore these are represented by some numbers followed by certain number of zeros. For example, the age of the universe is approximated to the . 8,000,000,000 years and the mass of the Earth is approximately 598000000000000000 metric tons. Usually such numbers are written by using exponents. For example, the number 8000000000 can be written as 8 × 109 or 80 × 108 or 800 × 107. Similarly, 598000000000000000 = 598 × 1019. This is known as exponential notation. Using exponential notation we can write very large and very small numbers easily.
Exponential Form:
Consider the number 125 we write 125 = 5 × 5 × 5. Therefore 125 = 53, 53 is the exponential form of 125. Here ‘5’ is base and ‘3’ is exponent. The number 53 is read as 5 raised to power of 3 or simply 5 cubed.
Consider another number \(\frac{81}{16}\)
\(\frac{81}{16}=\frac{3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2}=\left(\frac{3}{2}\right)^{4}\)
Therefore, the exponential form of \(\frac{81}{16}\) is \(\left(\frac{3}{2}\right)^{4}\)
Here \(\frac{3}{2}\) is base and 4 is exponent.
Therefore, if a is any rational number of n is a natural number then an = a × a × a …. multiplied n times, where a is called the base and n is called the exponent or index and an is the exponential form, an is read as a raised to the power n. In particular, a1 = a.
For example: 105 = 10 × 10 × 10 × 10 × 10 i.e. 105 = 100000 here base = 10, exponent (or index) = 5 and 105 is the exponential form of the number 100000.
Laws of Exponents
We can multiply and divide rational numbers expressed in exponential form.
Multiplication of Identical bases with different powers
Law 1: If a is any rational number and m, n are integers, then
am x an = am+n
Division of Identical bases with different powers
Law 2: If a is any (non zero) rational number and m, n are integers such that m > n, then
am ÷ an = \(\frac{a^{m}}{a^{n}}\) = am-n
Zero Exponent
Law 3: If a is any (non zero) rational number, then
a0 = 1
Taking power of a power
Law 4: If a is any rational number and m, n are integers, then
(am)n = am×n
Multiply different bases with same exponent
Law 5: If a, b are any rational number and n is an integers, then
an × bn – (ab)n
Division of different bases with same exponent
Law 6: If a, (b ≠ 0) are any numbers and n is an integer, then \(\frac{a^{n}}{b^{n}}=\left(\frac{a}{b}\right)^{n}\)
or
an ÷ bn = \(\left(\frac{a}{b}\right)^{n}\)
Negative Exponent
For any (non-zero) rational number a and natural number n, we have
\(\frac{1}{a^{n}}=\frac{a^{0}}{a^{n}}\) = a0-n = a-n, thus a-n = \(\frac{1}{a^{n}}\)
Law 7: If a is any (non-zero) rational number and n is an integer, then a-n = \(\frac{1}{a^{n}}\) In particular, a-1 = \(\frac{1}{a}\).
Decimal Number System
Consider the expansion of the number 863015, we know that
863015 = 8 × 100000 + 6 × 10000 + 3 × 1000 + 0 × 100 + 1 × 10 + 5 × 1
Using power of 10 in the exponent, we can write it as
863015 = 8 × 105 + 6 × 104 + 3 × 103 + 0 × 102+ 1 × 101 + 5 × 100
= 8 × 105 + 6 × 104 + 3 × 103 + 1 × 101 + 5 × 100
In fact, the expansion of every number can be written using power of 10 in the exponent.
Standard form of Numbers
The standard form of a number in of the form k × 10n where k is a number between 1 and 10 and n is an integer.
Look at the following
96 = 9.6 × 10 = 9.6 × 101
963 = 9.63 × 100 = 9.63 × 102
9630 = 9.63 × 1000 = 9.63 × 103
96300= 9.63 × 1000 = 9.63 × 104 and so on.
Scientific Notation
Scientific notation is a way of writing numbers that accommodates value too large to be conveniently written in decimal notation.
In scientific notation all numbers are written as k × 10n where k is decimal number such that 1 < k < 10 and n is a whole number. The decimal k is called significant. Scientific notation is also known as standard form.
| Ordinary decimal notation | Scientific notation |
| 600 | 6 × 102 |
| 57,000 | 5.7 × 104 |
| 6,830,000,000 | 6.83 × 109 |