In mathematics, two or more exponential terms which contain different bases and same powers are participated in multiplication. The product of them cannot be calculated directly but it can be done by applying the concept of exponentiation.

Look at the following examples to learn how to multiply the indices with same powers and different bases for beginners.

01

$(1) \,\,\,\,\,\,$ $2^3 \times 5^3$

According to exponentiation, write each term as the factors of its base.

$\,=\, $ $(2 \times 2 \times 2) \times (5 \times 5 \times 5)$

$\,=\, $ $2 \times 2 \times 2 \times 5 \times 5 \times 5$

$\,=\, $ $2 \times 5 \times 2 \times 5 \times 2 \times 5$

$\,=\, $ $(2 \times 5) \times (2 \times 5) \times (2 \times 5)$

Express the product of the factors in exponential form.

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

$\,=\, $ $10^3$

$\, \therefore \,\,\, 2^3 \times 5^3 \,=\, 10^3$

It is proved in this example that the product of exponential terms which have different bases and same exponents is equal to the product of the bases raised to the power of same exponent.

$\, \therefore \,\,\, 2^3 \times 5^3 \,=\, {(2 \times 5)}^3 \,=\, 10^3$

Observe the following exponents to understand how to multiply exponents with different bases and same powers.

$(2) \,\,\,\,\,\,$ ${(-3)}^5 \times 4^5 \,=\, {(-12)}^5$

$(3) \,\,\,\,\,\,$ ${(0.2)}^4 \times {(0.3)}^4 \,=\, {(0.06)}^4$

$(4) \,\,\,\,\,\,$ ${\Bigg(\dfrac{2}{3}\Bigg)}^{20} \times {\Bigg(\dfrac{5}{7}\Bigg)}^{20} \,=\, {\Bigg(\dfrac{10}{21}\Bigg)}^{20}$

$(5) \,\,\,\,\,\,$ ${(\sqrt{6})}^7 \times 4^7 \,=\, {(4\sqrt{6})}^7$

02

Apply the same fundamental procedure to learn how to multiply three exponential terms having same exponents and different bases to get product of them in same form.

$(1) \,\,\,\,\,\,$ $2^5 \times 3^5 \times 4^5$

$\,=\,$ $(2 \times 2 \times 2 \times 2 \times 2)$ $\times$ $(3 \times 3 \times 3 \times 3 \times 3)$ $\times$ $(4 \times 4 \times 4 \times 4 \times 4)$

$\,=\,$ $2 \times 2 \times 2 \times 2 \times 2$ $\times$ $3 \times 3 \times 3 \times 3 \times 3$ $\times$ $4 \times 4 \times 4 \times 4 \times 4$

$\,=\,$ $(2 \times 3 \times 4)$ $\times$ $(2 \times 3 \times 4)$ $\times$ $(2 \times 3 \times 4)$ $\times$ $(2 \times 3 \times 4)$ $\times$ $(2 \times 3 \times 4)$

$\,=\,$ ${(2 \times 3 \times 4)}^5$

$\,=\,$ $24^5$

$\,\, \therefore \,\,\,\,\,\,$ $2^5 \times 3^5 \times 4^5$ $\,=\,$ ${(2 \times 3 \times 4)}^5$ $\,=\,$ $24^5$

Observe the below best examples to understand the multiplication of exponential terms having different bases and same exponents.

$(2) \,\,\,\,\,\,$ ${(-2)}^{10} \times {(-3)}^{10} \times {(-4)}^{10} \,=\, {(-24)}^{10}$

$(3) \,\,\,\,\,\,$ ${(0.11)}^5 \times {(0.12)}^5 \times {(0.13)}^5 \,=\, {(0.014916)}^5$

$(4) \,\,\,\,\,\,$ ${\Bigg(\dfrac{2}{3}\Bigg)}^7 \times {\Bigg(\dfrac{4}{5}\Bigg)}^7 \times {\Bigg(\dfrac{6}{9}\Bigg)}^7 \,=\, {\Bigg(\dfrac{48}{135}\Bigg)}^7$

$(5) \,\,\,\,\,\,$ ${(\sqrt{6})}^4 \times {(\sqrt{7})}^4 \times {(\sqrt{8})}^4 \,=\, {(\sqrt{336})}^4$

Use the same fundamental procedure to multiply any number of exponents which have different bases but the exponent should be same in the terms.

The power of a product rule is derived in general algebraic form on the basis of the multiplication of exponents which have same power but different bases.