Exponents and Powers Class 7 Notes Maths Chapter 11

Students can go through AP 7th Class Maths Notes Chapter 11 Exponents and Powers to understand and remember the concepts easily.

Class 7 Maths Chapter 11 Notes Exponents and Powers

→ x + x + x + x + x + ………….. 20 times – 20x

→ x × x × x × x × x × ………….. 20 times = x²⁰

→ x + x + x + x + x + …………. n times = nx

→ x × x × x × x × x × …………… n times – xⁿ

→ The exponential form of 10,000 is 10⁴.

→ 10⁴ can be read as fourth power of 10 (or) 10 power 4.

→ 1.00,000 – 10⁵ = 10 × 10 × 10 × 10 × 10.

→ In 10⁶, 10 is called base and 6 is called exponent.

→ The number 10⁹ is read as 10 raised to the power of 9.

→ 81 = 3 × 3 × 3 × 3 = 34

→ 625 = 5 × 5 × 5 × 5 = 5⁴

→ 243 = 3 × 3 × 3 × 3 × 3 = 3⁵

→ a × a × a × a = a⁴

→ a × a × a × a × ………… × a …………. 10 times = a¹⁰

→ a × a × b × b = a²b²

→ a × a × a × b × b × b = a³b³

→ a × b × a × b × a × b × a × b = (a × b)⁴

→ a² mean a is squared

→ a³ mean a is cubed.

→ (- 1) odd number = – 1
Ex : 1) (- 1)³ = – 1 × – 1 × – 1 = -1
2) (- 1 )⁵ = – 1 × – 1 × – 1 × – 1 × – 1
= – 1

→ (-) Even number = 1
Ex : 1) (- 1)² = – 1 × – 1 = 1
2) (- 1)⁴ = – 1 × – 1 × – 1 × – 1 = 1

→ Law of Exponents
i) a^m × aⁿ = a^{m + n}
Ex:
1) 2³ × 2⁴ = 2^{3 – 4} = 2⁷
2) 4³ × 4⁷ = 4^{3 – 7} = 4¹¹

Note :
1) a^m × aⁿ × a^p = a^{m + n + p}
2) a^p × a^q × a^r × a^s = a^{p + q + r + s}
3) a^m × a^-n = a^{m – n}
4) a^-m × a^-n = a^-m-n

ii) a^m ÷ aⁿ = a^{m – n} (or) \(\frac{a^m}{a^n}\) = a^{m – n}
Ex:
1) 7¹² ÷ 7⁴ = 7^{12 – 4} = 7⁸
2) 8⁹ ÷ 8² = 8^{9 – 2} = 8⁷

iii) (a^m)ⁿ = a^mn
Ex:
1) (2³)⁴ = 2^{3 × 4} = 2¹²
2) (3²)² = 3^{2 × 2} = 3⁴

iv) a^m × b^m = (a × b)^m = (ab)^m
Ex : 1) 2³ × 12³ = (2 × 12)³ = 24³
2) 7⁴ × 6⁴ = (7 × 6)⁴ = 42⁴

v) a^m ÷ b^m = \(\frac{a^m}{b^m}\) = (\(\frac{\mathrm{a}}{\mathrm{~b}}\))^m
Ex : 1) 7⁴ ÷ 3⁴ = (\(\frac{7}{3}\))⁴
2) 4⁹ ÷ 7⁹ = (\(\frac{4}{7}\))⁹

v) a⁰, if a = 0
Ex :
1) 2⁰ = 1
2) 13⁰ = 1
3) (2024)⁰ = 1
Note :
1) a^-n = \(\frac{1}{a^n}\)
2) a^-1 = \(\frac{1}{\mathrm{a}}\)
3) (ab)^-1 = \(\frac{1}{a b}\).
4) \(\frac{(\mathrm{ab})^{\mathrm{m}}}{\mathrm{c}^{\mathrm{n}}}\) = a^mb^m × c^-n
5) aⁿ = \(\frac{1}{a^{-n}}\)

→ Observe the following.
59 = 5.9 × 10 = 5.9 × 10¹
590 = 5.9 × 100 = 5.9 × 10²
5900 = 5.9 ×1000 = 5.9 × 10³
59000 = 5.9 × 10000 = 5.9 × 10⁴

→ Any number can be expressed as a decimal number between 1.0 and 10.0 including 1.0 multiplied by a power of 10 such a form of a number is called its standard form.
Ex:
1) 6,985 = 6.985 × 1000
= 6.985 × 10³
2) 231 = 2.31 × 100
= 2.31 × 10²