CBSE Notes Class 7 Chapter 11 Exponents and Powers

Chapter 11 of CBSE Class 7 Mathematics, "Exponents and Powers," introduces students to the concise representation of large numbers using exponents. It explores concepts such as base and exponent, simplifying expressions using laws of exponents, and representing numbers in standard form. This chapter helps students understand operations involving powers, including multiplication and division, and using negative exponents. It lays a strong foundation for handling larger numerical values efficiently in further mathematical studies and practical applications.

1.0Exponent Form of a Number

Exponent form is used to express large numbers in a compact and readable way. For example, the mass of an electron is $9.1\times 10^{-31}$ kg, and the world population is around $7\times 10^9$. In exponent notation, if $x$ is a rational number, $x^n$ represents the product of x multiplied by itself n times, where x is the base and n is the exponent. 

For example, $7\times 7\times 7\times 7=7^4$. If the base is negative, the rule remains the same, but the sign depends on whether the exponent is odd (negative value) or even (positive value).

Exponents are shorthand for repeated multiplication, and expressions like $a^3{b}^2$ (read as "a cubed b squared") represent repeated factors. Additionally, the order of multiplication does not affect the result, so $a^5{b}^3$ is the same as $b^3{a}^5$.

2.0Laws of Exponents

1. Product of Powers Rule: $a^m\times a^n=a^{m+n}$
2. Power of a Power Rule: $(am)n=a^{mn}$
3. Power of a Product Rule: $(ab{)}^n=a^n\times b^n$
4. Zero Exponent Rule: $a^0=1$
5. Negative Exponent Rule: $a^{-n}=\frac{1}{a^n}$
6. Quotient Law of Exponent: $\frac{a^m}{a^n}=a^{m-n}$

These rules help simplify and solve expressions involving exponents.

3.0Writing Expanded Form of a Decimal Number using Exponential Notation

The expanded form of the decimal number 35462.963 can be written as:

$35462.963=3\times 10^4+5\times 10^3+4\times 10^2+6\times 10^1+2\times 10^0+9\times 10^{-1}+6\times 10^{-2}+3\times 10^{-3}$

In this notation, each digit is multiplied by a power of 10, with the exponents decreasing by 1 from left to right.

4.0Scientific Notation

Scientific notation expresses large or small numbers in the form $k\times 10^n$, where $1\le k<10$ and n is an integer. This is also known as standard form. 

For example:

• The Sun's distance from the center of the Milky Way is $3.0\times 10^{20}$ meters.
• The size of an atom is $3.0\times 10^{-8}$ centimeters.

This notation simplifies writing extremely large or small numbers.

5.0Solved Example of Exponents and Powers

Example 1: Find the value of (–10)⁴

Solution: (–10)⁴ = –10 × –10 × –10 × –10 = 10000

Example 2: Simplify  $\frac{-27}{125}=-\frac{27}{125}$

Solution: $\frac{-27}{125}=\frac{-3\times -3\times -3}{5\times 5\times 5}=(\frac{-3}{5}{)}^3$

Example 3: Simplify $(\frac{-9}{7}{)}^4\times (\frac{-9}{7}{)}^{-5}$

Solution:$(\frac{-9}{7}{)}^4\times (\frac{-9}{7}{)}^{-5}=(\frac{-9}{7}{)}^{4+(-5)}(\frac{-9}{7}{)}^{4-5}=(\frac{-9}{7}{)}^{-1}$

Example 4: Simplify $((-\frac{1}{3}{)}^{-3}\times \frac{4}{5}{)}^5$

Solution:

$((-\frac{1}{3}{)}^{-3}\times \frac{4}{5}{)}^5=((-\frac{1}{3}{)}^{-15}\times (\frac{4}{5}{)}^5))$

Example 5: Find the values of $(\frac{x^2}{3y^3}{)}^0$

Solution:

$(\frac{x^2}{3y^3}{)}^0=1$

Example 6: Simplify $((\frac{(-2)(3)}{(1)(-5)}{)}^3$

Solution:

$((\frac{(-2)(3)}{(1)(-5)}{)}^3=(\frac{-6}{-5}{)}^3=(\frac{6}{5}{)}^3=(\frac{6\times 6\times 6}{5\times 5\times 5})=\frac{216}{125}$

Example 7: Find the value of the following using laws of exponents.

$\frac{2^8\times 3^4}{8\times (-2{)}^5\times 27}$

Solution:

$\frac{2^8\times 3^4}{2^3\times (-1{)}^5(2{)}^5\times 3^3}=\frac{2^{8-3-5}\times 3^{4-3}}{(-1)}$

$-1\times 2^0\times 3=-1\times 3=-3(2^0=1)$

6.0Sample Questions on Exponents and Powers

1. What is an exponent?  

Ans: An exponent is the number that indicates how many times the base number is multiplied by itself. For example, in $2^3$, 2 is the base, and 3 is the exponent, meaning $2\times 2\times 2=8$.

2. What is the meaning of negative exponents?  

Ans: A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For example, $2^{-3}=\frac{1}{2^3}=\frac{1}{8}$.

3. What is the rule for multiplying powers with the same base?  

Ans: When multiplying powers with the same base, add the exponents:  

$a^m\times a^n=a^{m+n}$

4. What is the rule for dividing powers with the same base?  

Ans: When dividing powers with the same base, subtract the exponents:  

$\frac{a^m}{a^n}=a^{m-n}$

5. What happens when the exponent is 0?  

Ans: Any non-zero number raised to the power of 0 is 1:  $a^0=1(fora\ne 0)$

6. What is scientific notation?  

Ans: Scientific notation is a way of expressing very large or small numbers in the form $k\times 10^n$, where $1\le k<10$ and n is an integer. For example, $3.0\times 10^8$ represents 300,000,000.

7. How do you express numbers with fractional exponents?  

Ans: A fractional exponent represents both a root and a power. For example, $a^{\frac{m}{n}}=\sqrt[n]{a^m}$, which means the n^th root of a raised to the power m.

8. What is the difference between powers and exponents?  

Ans: "Exponent" refers to the number that indicates how many times to multiply the base. "Power" refers to the result of raising the base to the exponent. For example, in $2^3$, 3 is the exponent, and 8 is the power.