The exponents, which have the same base participate in the multiplication and the product of them should be evaluated in mathematics. Just like multiplying the numbers, it is not correct to multiply a quantity in exponential form by another quantity in exponential notation.

Now, let’s learn how to multiply two or more exponents whose bases are same, from some examples.

$2^3 \times 2^4$

Look at the expression in arithmetic form.

- Two quantities in exponential notation are involved in the multiplication.
- The numbers $3$ and $4$ are the exponents in the expression.
- The exponents $3$ and $4$ both have same bases and it is $2$ in this example.

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.

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