Two or more powers with the same base often participate in multiplication in mathematics. It is not possible to multiply the exponents with same base directly but the concept of exponentiation is applied while performing the multiplication of exponents with the same base for obtaining product of them.

Here is the step by step procedure for beginners and it helps you to do multiplication of indices with the same base.

01

$(1) \,\,\,\,\,\,$ $2^3 \times 2^4$

According to exponentiation, write each term as the factors of $2$.

$\,=\, $ $(2 \times 2 \times 2) \times (2 \times 2 \times 2 \times 2)$

$\,=\, $ $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

Write the product in exponential notation.

$\,=\, $ $2^7$

$\, \therefore \,\,\, 2^3 \times 2^4 \,=\, 2^7$

You can observe in this example that the exponent of product of exponents with same base is equal to the summation of the exponents.

$\, \therefore \,\,\, 2^3 \times 2^4 \,=\, 2^{3+4} \,=\, 2^7$

You can multiply the powers with the same in this way and repeat the same procedure for the following examples.

$(2) \,\,\,\,\,\,$ ${(-3)}^7 \times {(-3)}^5 \,=\, {(-3)}^{12}$

$(3) \,\,\,\,\,\,$ ${(0.7)}^3 \times {(0.7)}^3 \,=\, {(0.7)}^6$

$(4) \,\,\,\,\,\,$ ${\Bigg(\dfrac{2}{3}\Bigg)}^{11} \times {\Bigg(\dfrac{2}{3}\Bigg)}^{19} \,=\, {\Bigg(\dfrac{2}{3}\Bigg)}^{30}$

$(5) \,\,\,\,\,\,$ ${(\sqrt{6})}^5 \times {(\sqrt{6})}^8 \,=\, {(\sqrt{6})}^{13}$

02

Apply same procedure for multiplication of three terms to obtain product of them.

$(1) \,\,\,\,\,\,$ $9^3 \times 9^4 \times 9^5$

$\,=\,$ $(9 \times 9 \times 9)$ $\times$ $(9 \times 9 \times 9 \times 9)$ $\times$ $(9 \times 9 \times 9 \times 9 \times 9)$

$\,=\,$ $9 \times 9 \times 9$ $\times$ $9 \times 9 \times 9 \times 9$ $\times$ $9 \times 9 \times 9 \times 9 \times 9$

$\,=\,$ $9^{12}$

$\,\, \therefore \,\,\,\,\,\,$ $9^3 \times 9^4 \times 9^5$ $\,=\,$ $9^{3+4+5}$ $\,=\,$ $9^{12}$

Observe the following examples to learn how to multiply three exponents with the same base.

$(2) \,\,\,\,\,\,$ ${(-4)}^2 \times {(-4)}^3 \times {(-4)}^4 \,=\, {(-4)}^9$

$(3) \,\,\,\,\,\,$ ${(0.12)}^4 \times {(0.12)}^3 \times {(0.12)}^4 \,=\, {(0.12)}^{11}$

$(4) \,\,\,\,\,\,$ ${\Bigg(\dfrac{1}{4}\Bigg)}^6 \times {\Bigg(\dfrac{1}{4}\Bigg)}^7 \times {\Bigg(\dfrac{1}{4}\Bigg)}^8 \,=\, {\Bigg(\dfrac{1}{4}\Bigg)}^{21}$

$(5) \,\,\,\,\,\,$ $\sqrt{2} \times {(\sqrt{2})}^3 \times {(\sqrt{2})}^5 \,=\, {(\sqrt{2})}^9$

The same mathematical approach is applied to any number of exponents with the same base to multiply them and also to get product of them easily.

Add exponents and write sum of them as exponent of the base when two or more exponents with the same base are multiplied in mathematics. On the basis of this, adding exponents product rule is developed in general form to use it as a formula in mathematics.

Latest Math Topics

Mar 21, 2023

Feb 25, 2023

Feb 17, 2023

Feb 10, 2023

Jan 15, 2023

Latest Math Problems

Mar 03, 2023

Mar 01, 2023

Feb 27, 2023

A best free mathematics education website for students, teachers and researchers.

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Learn how to solve the math problems in different methods with understandable steps and worksheets on every concept for your practice.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved