Multiplying exponents with the same base

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.

Examples

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

Multiplying Two Exponential terms

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

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.

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

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

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

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

Multiplying Three Exponential terms

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.