Multiplying Powers With the Same Base

Exponent multiplication is the process of multiplying two exponent-containing expressions. Depending on the base and the power, specific rules apply when multiplying exponents. Different bases, negative exponents, and non-integer exponents can occasionally make it challenging for learners to understand. In this tutorial, let’s study more about multiplying exponents.

In short, multiplying powers or exponents with the same base implies that the different exponents must be multiplied by each other in order to get the answer. Here, an example is given for your reference:
2³*2⁴= 2³⁺⁴ =2⁷= 128

Wondering how this happens? There are certain rules that govern the exponential powers or bases. Continue reading the article further to know the details.

KNOW MORE ABOUT EXPONENTS AND POWERS

Rules for Multiplying Exponents with the Same Base of Exponents

Following are the seven laws of exponents that you must know:

Law 1: Multiplication of powers with a common base
The general form of this law is ?^?×?^?=?^?+n
Law 2: Dividing exponents with the same base
The general form of this law is ?^?÷?^?=?^?–?
Law 3: The law of the power of a power
The general form of this law is ?^??=?^?×?
Law 4: The law of multiplication of powers with different bases but same exponents.
The general form of this law is ?^?×?^?=(?×?)^?
Law 5: The law of the division of powers with different bases but the same exponents.
The general form of this law is ?^?÷?^?=?^? /?^?=(?/?)^?
Law 6: The law of negative exponents.
The general form of this law is ?^–?=1/aⁿ
Law 7: Zero power rule
The general form of this law is a⁰=1.

Rules for Multiplying Exponents with the Same Base

Consider two numbers or expressions having the same base, that is, aⁿ and a^m. Here, the base is ‘a’. When the terms with the same base are multiplied, the powers are added, i.e., a^m × aⁿ = a^{m+n}

Let us explore some examples to understand how the powers are added.

Example 1: Multiply 2³ × 2²

Solution: Here, the base is the same, that is, 2. According to the rule, we will add the powers, 2³ × 2² = 2⁽³⁺²⁾ = 2⁵= 32.
Let us verify the answer. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 2 × 2 × 2 × 2 × 2 = 2⁵ = 32

Example 2: Find the product of 10⁴⁵ and 10³⁹

Solution: In the given question, the base is the same, that is, 10. According to the rule, we will add the powers, 10⁴⁵ × 10³⁹ = 10^(45+39) = 10⁸⁴.

Detailed formula-wise rules are given here in the table below:

Exponential Forms	Rules
When the bases are the same.	a^-n × a^-m= a^-(n+m)= 1/a^{n+m}
When the bases are different and the negative powers are the same.	a^-n× b^-n= (a × b)^-n = 1/(a × b)ⁿ
When the bases and the negative powers are different.	a^-n × b^-m= (a^-n) × (b^-m)

Multiplying Fractional Exponents Having Same Base

A fractional exponent is an expression that has a fractional power. An illustration of a fractional exponent is 23/5. With the aid of the following table, let’s learn the guidelines for multiplying fractional exponents.

Exponential Forms	Rules
When the bases are the same.	a^n/m× a^k/j= a^n/m+k/j
When the bases are different but the fractional powers are the same.	a^n/m× b^n/m= (a×b)^n/m
When the bases and the fractional powers are different.	a^n/m× b^k/j= (a^n/m) × (b^k/j)

Problems on Exponents

Evaluate the following questions given on multiplying powers with the same base:

4⁻³
(12)^-5
(43)^-3
(-3)^-4
(-23)^-5
(2⁰ + 3⁻¹) × 3²
(2⁻¹ × 3⁻¹) ÷ 2⁻³

Hope this comprehensive article on “Multiplying Powers with the same base” helps you in your Mathematics exam preparation. Stay tuned to embibe.com for more updates!