How do you deal with negative bases in exponents?

The sign of the result depends on whether the exponent is odd or even: - If the exponent is odd, the result is negative. - If the exponent is even, the result is positive.

Chapter 11 Exponents And Powers

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CBSE Notes Class 7 Chapter 11 Exponents and Powers

Q: What is the difference between powers and exponents?

"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.

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 $9.1 \times 1 0^{- 31}$ kg, and the world population is around $7 \times 1 0^{9}$ . 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

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

These rules help simplify and solve expressions involving exponents.

Also Read: Laws of 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:

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

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 $k \times 1 0^{n}$ , where $1 \leq k < 10$ 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 $3.0 \times 1 0^{20}$ meters.
The size of an atom is $3.0 \times 1 0^{- 8}$ 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 $(\frac{x ^{2}}{3 y ^{3}})^{0}$

Solution:

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

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

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 $2 \times 2 \times 2 = 8$ .

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, $2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ .

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

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

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

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

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

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

What happens when the exponent is 0?

Ans: Any non-zero number raised to the power of 0 is 1: $a^{0} = 1 (f or a \neq = 0)$

What is scientific notation?

Ans: Scientific notation is a way of expressing very large or small numbers in the form $k \times 1 0^{n}$ , where $1 \leq k < 10$ and n is an integer. For example, $3.0 \times 1 0^{8}$ represents 300,000,000.

How do you express numbers with fractional exponents?

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

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.

7.0Key Features of Class 7 Chapter 11 : Exponents and Powers

Introduction to Exponents: Simple explanation of powers, bases, and exponents with everyday examples.
Laws of Exponents: Clear and detailed coverage of the important laws, like multiplication, division, and power of a power.
Negative Exponents: Easy-to-understand rules for dealing with negative powers.
Standard Form: Explanation of expressing large and small numbers in standard form using exponents.
Use of Exponents in Real Life: Practical applications of exponents in scientific and mathematical contexts.
Solved Examples: Step-by-step examples to demonstrate the correct use of exponent rules.
Practice Problems: Varied exercises to develop problem-solving skills and strengthen conceptual clarity.

Chapter-wise CBSE Notes for Class 7 Maths:

Class 7 Maths Chapter 1 - Integers Notes

Class 7 Maths Chapter 2 - Fractions and Decimals Notes

Class 7 Maths Chapter 3 - Data Handling Notes

Class 7 Maths Chapter 4 - Simple Equations Notes

Class 7 Maths Chapter 5 - Lines And Angles Notes

Class 7 Maths Chapter 6 - The Triangles and its Properties Notes

Class 7 Maths Chapter 7 - Comparing Quantities Notes

Class 7 Maths Chapter 8 - Rational Numbers Notes

Class 7 Maths Chapter 9 - Perimeter And Area Notes

Class 7 Maths Chapter 10 - Algebraic Expressions Notes

Class 7 Maths Chapter 11 - Exponents And Powers Notes

Class 7 Maths Chapter 12 - Symmetry Notes

Class 7 Maths Chapter 13 - Visualising Solid Shapes Notes

Chapter-wise NCERT Solutions for Class 7 Maths:-

Chapter 1: Integers

Chapter 2: Fractions and Decimals

Chapter 3: Data Handling

Chapter 4: Simple Equations

Chapter 5: Lines and Angles

Chapter 6: The Triangle and its Properties

Chapter 7: Comparing Quantities

Chapter 8: Rational Numbers

Chapter 9: Perimeter and Area

Chapter 10: Algebraic Expressions

Chapter 11: Exponents and Powers

Chapter 12: Symmetry

Chapter 13: Visualising Solid Shapes

Frequently Asked Questions

CBSE Class 7

The CBSE Class 7 exam is an important milestone in a student's educational journey and is conducted under the Central Board of Secondary Education guidelines...

CBSE Class 7 Syllabus

The CBSE Class 7 syllabus is crucial for organized learning and exam preparation, providing a clear understanding of...

CBSE Class 7 Notes

CBSE Class 7 Notes are important for understanding the basic concepts in all subjects and building a strong foundation for advance classes.

NCERT Solutions Class 7

The questions and illustrations in the NCERT Class 7 Science textbook have comprehensive explanations and answers...

CBSE Exam

CBSE is a national-level board of education in India, responsible for public and private school education, and is managed by the Gove...

Important CBSE Science Topics

Science is, therefore, a systematic enterprise that organizes and builds testable explanations and predic...

Important CBSE Maths Topics

Mathematics is the study of patterns, structure, and relationships, rooted in fundamental practices like counting,...

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I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

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 $9.1 \times 1 0^{- 31}$ kg, and the world population is around $7 \times 1 0^{9}$ . 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

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

These rules help simplify and solve expressions involving exponents.

Also Read: Laws of 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:

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

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 $k \times 1 0^{n}$ , where $1 \leq k < 10$ 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 $3.0 \times 1 0^{20}$ meters.
The size of an atom is $3.0 \times 1 0^{- 8}$ 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 $(\frac{x ^{2}}{3 y ^{3}})^{0}$

Solution:

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

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

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 $2 \times 2 \times 2 = 8$ .

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, $2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}$ .

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

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

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

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

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

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

What happens when the exponent is 0?

Ans: Any non-zero number raised to the power of 0 is 1: $a^{0} = 1 (f or a \neq = 0)$

What is scientific notation?

Ans: Scientific notation is a way of expressing very large or small numbers in the form $k \times 1 0^{n}$ , where $1 \leq k < 10$ and n is an integer. For example, $3.0 \times 1 0^{8}$ represents 300,000,000.

How do you express numbers with fractional exponents?

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

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.

7.0Key Features of Class 7 Chapter 11 : Exponents and Powers

Introduction to Exponents: Simple explanation of powers, bases, and exponents with everyday examples.
Laws of Exponents: Clear and detailed coverage of the important laws, like multiplication, division, and power of a power.
Negative Exponents: Easy-to-understand rules for dealing with negative powers.
Standard Form: Explanation of expressing large and small numbers in standard form using exponents.
Use of Exponents in Real Life: Practical applications of exponents in scientific and mathematical contexts.
Solved Examples: Step-by-step examples to demonstrate the correct use of exponent rules.
Practice Problems: Varied exercises to develop problem-solving skills and strengthen conceptual clarity.