Operations with powers

Let's now see how to manage the common operations with powers:

Powers product

If we want to do the product of two powers with the same base, for example $4^3$ and $4^6$, we will do the following:$$\begin{array}{rcl}{4}^3& =& 4\cdot 4\cdot 4\\ 4^6& =& 4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4\end{array}$$Therefore, if we multiply them we will have: $$4^3\cdot 4^6=(4\cdot 4\cdot 4)\cdot (4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4)=4^9=4^{3+6}$$In general, if we want to do the product of two powers with the same base, the result is a power with the same base, but the exponent is the sum of the exponents. This is: $a^n\cdot a^m=a^{n+m}$.

Power quotient:

Similar to the product, it is possible to calculate the ratio of two powers of the same base. For example:$${\displaystyle \frac{4^6}{4^2}=\frac{4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}=4\cdot 4\cdot 4\cdot 4=4^{6-2}}$$ Therefore we say that, in general, the procedure is:${\displaystyle \frac{a^n}{a^m}=a^{n-m}}$. Let's illustrate this in some calculations:

Power of a product:

If we want to do the following operation $(3\cdot 4{)}^3$ we observe that $(3\cdot 4{)}^3=(3\cdot 4)\cdot (3\cdot 4)\cdot (3\cdot 4)=(3\cdot 3\cdot 3)\cdot (4\cdot 4\cdot 4)=3^3\cdot 4^3$. To calculate the result we have two options, either multiply 3 by 4 and cube the product (raise to the power three) : $(3\cdot 4{)}^3=(12{)}^3=1728$ or cube each of the factors of the product: $3^3=27$ and $4^3=64$, therefore the product will be $27\cdot 64=1728$. In general, then, the power of a product is equal to the product of the power. That is, $(a\cdot b{)}^m=a^m\cdot b^m$. For instance:

Power of a ratio:

Similar to the product, in general ${\displaystyle (\frac{a}{b}{)}^m=\frac{a^m}{b^m}}$. Let's see some examples:

Power of a power:

To calculate expressions like, for example $(2^3{)}^5$, we can also see it as: $(2^3{)}^5=(2^3)\cdot (2^3)\cdot (2^3)\cdot (2^3)\cdot (2^3)=2^{3+3+3+3+3}=2^{15}$ Therefore we realize that in doing a power of a power, the exponents are multiplied and the base is the same. In general, this is expressed as: $(a^n{)}^m=a^{n\cdot m}$ Let's see some examples:

Using all these properties of the powers, we can now work with more comfortable expressions when we need to calculate the product of a number by itself many times. To assimilate completely the concept, let's see a few examples that use all the properties: