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: $$a^n\cdot a^m=a^{n+m}$$.

$$$\displaystyle 2^4\cdot 2^5= 2^{4+5}=2^9 \\ 9^{21}\cdot 9^5 = 9^{21+5}=9^{26}$$$

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:

$$\displaystyle \frac{3^7}{3^3}=\frac{3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 3}=3 \cdot 3 \cdot 3 \cdot 3=3^4=3^{7-3}$$

$$\displaystyle \frac{17^{34}}{17^{12}}=17^{34-12}=17^{22}$$

$$\displaystyle \frac{4^3}{4^3}=4^{3-3}=4^0=1$$

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:

$$\displaystyle (4 \cdot 2 \cdot 10)^2=4^2\cdot 2^2\cdot 10^2=16 \cdot 5 \cdot 100 =6400$$

$$\displaystyle (3\cdot 5)^3=3^3\cdot 5^3=27 \cdot 625 = 16875$$

Power of a ratio:

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

$$$\displaystyle \Bigg(\frac{6}{7}\Bigg)^2= \frac{6^2}{7^2}=\frac{36}{49} \\ \Bigg(\frac{5}{21}\Bigg)^8 = \frac{5^8}{21^8}=\frac{390625}{37822859361}$$$

Power of a power:

To calculate expressions like, for example $$\Big(2^3\Big)^5$$, we can also see it as: $$\Big(2^3\Big)^5=(2^3) \cdot (2^3) \cdot (2^3) \cdot (2^3) \cdot (2^3)=2^{3+3+3+3+3}=2^{15}$$ 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: $$\Big( a^n\Big)^m= a^{n\cdot m}$$ Let's see some examples:

$$$\displaystyle (4^2)^5= 4^{2 \cdot 5}= 4^{10} \\ (9^5)^7=9^{5\cdot 7}= 9^{35}$$$

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:

$$$\displaystyle 3^4\cdot 3^2- \displaystyle \frac{5\cdot (5^4)^2}{5^3}=3^{4+2}-\frac{5 \cdot 5^{4\cdot 2}}{5^3}=3^6-\frac{5 \cdot 5^8}{5^3}=3^6-\frac{5^{1+8}}{5^3}=3^6-\frac{5^9}{5^3}=3^6-5^{9-3}=3^6-5^6 \\ \displaystyle \frac{3 \cdot 6^{10} \cdot (3^4)^2\cdot 6}{3 \cdot 6}=\frac{3\cdot 3^{4\cdot 2}\cdot 6^{10+1}}{3 \cdot 6} = 3^{9-1}\cdot 6^{11-1}= 3^8\cdot 6^{10} \\ \displaystyle 4^{-2}=\frac{1}{4^2} \\ (3^7)^{-2}= 3^{7\cdot (-2)}=3^{-14} \\6^{-8}\cdot 6^3 = \displaystyle \frac{6^3}{6^8}= 6^{3-8}= 6^{-5}=\displaystyle \frac{1}{6^5}$$$