Class 7 - Exponents and Powers

Exponents and powers are fundamental concepts in mathematics that are used to represent repeated multiplication in a compact and efficient form. Exponents allow us to express large numbers in a simple and compact form, instead of writing the same number multiple times. These concepts are essential for solving problems in algebra, science, and everyday calculations. In this guide, you will learn the meaning of exponents and powers, their laws, and how to apply them step by step to simplify expressions and solve mathematical problems easily and accurately.

 

Table of Contents

• What are Exponents and Powers?
• Key Terminology: Base, Exponent, Power
• General Form of Exponents and Powers
• Laws of Exponents
• Solve Examples on Exponents and Powers
• Real-life applications of Exponents and Powers
• Common Mistakes to Avoid in Exponents and Powers

What are Exponents and Powers?

An exponent tells us how many times a number (called the base) is multiplied by itself. When you need to multiply the same number by itself, over and over again, writing it out every time gets tedious and error-prone very quickly. Use of exponents and powers solves this problem.
The shorthand of repeated multiplication is termed 'exponents'. The notation for writing the product of an integer by itself for a finite number of times is called the

exponential form. Observe the following example: 6 × 6 × 6 × 6. In this example, 6 is multiplied by itself 4 times. So, we write the expression in the exponential form as 6 × 6 × 6 × 6 = 6^{4} and read it as ‘6 to the power 4’. The raised number 4 is the exponent, and the whole expression 6^{4} is called the power.

 

Key Terminology: Base, Exponent, Power

Consider the expression  53

So,  53 = 5 × 5 × 5 = 125.

Consider 3 × 3 × 3 × 3 × 3 × 3 × 3; 3 is multiplied by itself 7 times. So, the exponential form of the expression is 37 , and it is read as '3 to the power of 7'.

 

In general, if we multiply ‘a’ ‘m’ times, i.e., a × a × a × a ... m times, it will be

represented as am (‘a to the power m’). Here ‘a’ is a rational number and ‘m’ is a natural number. In a^{m}, a is called the base, and m is called the exponent or index.

The base of am can also be negative.

For example, in (-4) × (-4) × (-4) = (-4)^{3}, the base is (-4) and the exponent is 3.

Laws of Exponents

The 7 laws of exponents are explained in this section:

Let ‘a’ be any number or integer (positive or negative) and ‘m’ and ‘n’ are positive integers, denoting the power to the bases, then;

• Product rule: The product of two exponents with the same base and different powers equals the base raised to the sum of the two powers or integers.
    am×an=am+n

• Quotient rule: When two exponents having the same base and different powers are divided, the result is the base raised to the difference between the two powers.
    am÷an=am−n (where a ≠ 0, m > n)

• Power of a Power Rule: When a power is raised to another power, multiply the exponents.
    (am)n=am×n

• Power of a Product Rule: A power distributed over a product, where each factor gets the exponent.
    (ab)m=am×bm

• Power of a Quotient Rule: A power applied to a fraction applies to both the numerator and denominator.
  (a/b)^{m} = a^{m}/ b^{m} (b ≠ 0)

• Zero Exponent Rule: Any non-zero number raised to the power of zero equals 1.
    a0=1 (where a ≠ 0)

• Negative Exponent Rule: A negative exponent means take the reciprocal of the base raised to the positive exponent.
    a−n=1/an (where a ≠ 0).
  
  

Solve Examples on Exponents and Powers

Example 1: Simplify: 45×43
Solution:  45×43=45+3=48

Example 2: Simplify:  (37)5×(37)7×(37)9
Solution:  (37)5×(37)7×(37)9=(37)5+7+9=(37)21

Example 3 : Simplify:  55÷53
Solution: 55÷53=55−3=52 . 

Example 4: Find the value of n , when  32n−1÷33=81
Solution:  32n−1÷33=81
⇒ 32n−1÷33=34
⇒  32n−1−3=34
⇒ 2n - 4 = 4
⇒ 2n = 8 ⇒ n = 4

Example 5: 53×p7×p3×q425×p5×q2
Solution: 53×p7×p3×q425×p5×q2

=53×p7+3×q425×p5×q2

=53×p10×q425×p5×q2

=5325×p10p5×q4q2

=53−2×p10−5×q4−2

=5×p5×q2

  =5p5q2

Example 6: If  xy=(35)2×(47)0 . Find the value of  (yx)2
Solution: xy=(35)2×(47)0
  xy=(35)2×1
  xy=3252=925
  yx=259
  (yx)2=(259)2=25292=62581

Real-life applications of Exponents and Powers

• Science (very large & very small numbers): Exponents help express large distances like 3 \times 10^8 m/s (speed of light) and very small sizes like atomic scales such as  10−10 m in a simple form.

• Computer Science (data & storage): Powers of 2 are used to measure memory and data sizes, such as  1KB=210 bytes and  1GB=230 bytes, making digital systems efficient to compute.

• Population Growth: Exponents model how populations grow exponentially over time when growth happens by a fixed percentage, showing a rapid increase in population in cities, bacteria, or species.

• Finance (Compound Interest): Exponents explain how money grows using formulas like  A=P(1+r)t , where returns increase on both the original amount and accumulated interest.

• Medicine & Biology: Exponential growth describes how bacteria and viruses multiply rapidly by doubling, helping in studying infections and the spread of diseases.

• Earth & Environment: Exponents are used in measuring earthquakes (Richter scale) and sound intensity (decibels), where small changes represent huge differences in energy.

• Everyday Life: Exponents help explain real-world patterns like social media growth, patterns of battery usage, and how small, consistent savings can grow significantly over time.

 

Common Mistakes to Avoid in Exponents and Powers

• Multiplying bases instead of adding exponents: A common mistake is treating exponent rules like normal multiplication instead of using  am×an=am+n .

• Adding exponents when bases are different: Exponents can only be added when the base is the same; different bases must be handled separately.

• Thinking a^0 = 0: Any non-zero number raised to the zero power equals 1, not 0.

• Forgetting the negative exponent rule: A negative exponent means reciprocal, like  a−n=1an​ , not a negative number.

• Confusing −32 and  (−(−3)2 : Without brackets,  −32=−(32) , but  (−3)2=9 .

• Using the product rule with different bases: Rules like a^m \times a^n work only when the bases are the same, not for different numbers.