Archivio Categoria: Proprietà delle potenze

Proprietà delle potenze


Prodotto di potenze con la stessa base

\displaystyle a^m \cdot a^n=a^{m+n}

Quoziente di potenze con la stessa base

\displaystyle\frac{a^m }{a^n}=a^{m-n}

Potenza di potenza

\displaystyle \left(a^m\right)^n=a^{m\cdot n}

Prodotto di potenze con lo stesso esponente

\displaystyle a^n \cdot b^n=(a\cdot b)^n

Quoziente di potenze con lo stesso esponente

\displaystyle\frac{a^n}{ b^n}=\left(\frac{a}{b}\right)^n

Potenza con esponente negativo

\displaystyle a^{-n}=\frac{1}{a^n}

\displaystyle \left(\frac{a}{ b}\right)^{-n}=\left(\frac{b}{ a}\right)^{n}

 ESEMPI

1) \displaystyle 2^3 \cdot 2^5=2^{3+5}=2^8

2) \displaystyle 2^7:2^5=2^{7-5}=2^{2}

3) \displaystyle \left(2^7\right)^3=2^{7\cdot 3}=2^{21}

4) \displaystyle 5^4 \cdot 2^4=(5\cdot 2)^4=10^4

5) \displaystyle 25^4:5^4=(25: 5)^4=5^4

6) \displaystyle \left(\frac{2}{3}\right)^3 \cdot \left(\frac{2}{3}\right)^5= \left(\frac{2}{3}\right)^{3+5}= \left(\frac{2}{3}\right)^8

7) \displaystyle \left(\frac{2}{3}\right)^7: \left(\frac{2}{3}\right)^5= \left(\frac{2}{3}\right)^{7-5}= \left(\frac{2}{3}\right)^{2}

8) \displaystyle \left[ \left(\frac{2}{3}\right)^7\right]^3= \left(\frac{2}{3}\right)^{7\cdot 3}= \left(\frac{2}{3}\right)^{21}

9) \displaystyle \left(\frac{2}{3}\right)^4 \cdot \left(\frac{3}{4}\right)^4= \left(\frac{2}{3}\cdot \frac{3}{4}\right)^4= \left(\frac{1}{2}\right)^4

10) \displaystyle \left(\frac{2}{3}\right)^4:\left(\frac{8}{3}\right)^4=\left(\frac{2}{3}\cdot \frac{3}{8}\right)^4=\left(\frac{1}{4}\right)^4

11) \displaystyle\frac{16^5}{ 8^5}=\left(\frac{16}{8}\right)^5=2^5

12) \displaystyle\left(\frac{5}{ 3}\right)^{-2}=\left(\frac{3}{ 5}\right)^{+2}=\frac{9}{25}

13) \displaystyle 2^7:2^9=2^{7-9}=2^{-2}=\left(\frac{1}{2}\right)^2

14) \displaystyle 2^{-2}\cdot2^{-3}=2^{-2-3}=2^{-5}=\left(\frac{1}{2}\right)^{5}

15) \displaystyle 2^{-2}:2^{-3}=2^{-2-(-3)}=2^{-2+3}=2

16) \displaystyle \left(-\frac{2}{3}\right)^{10} \cdot \left(+\frac{2}{3}\right)^8=\left(+\frac{2}{3}\right)^{10} \cdot \left(+\frac{2}{3}\right)^8=\left(+\frac{2}{3}\right)^{18}

17)  \displaystyle \left(-\frac{2}{3}\right)^{11} \cdot \left(+\frac{2}{3}\right)^8=-\left(+\frac{2}{3}\right)^{11} \cdot \left(+\frac{2}{3}\right)^8=-\left(+\frac{2}{3}\right)^{19}

18) \displaystyle \left(\frac{2}{3}\right)^{10}\cdot\left(\frac{3}{2}\right)^8=\left(\frac{2}{3}\right)^{10}\cdot\left(\frac{2}{3}\right)^{-8}=\left(\frac{2}{3}\right)^{10-8}=\left(\frac{2}{3}\right)^{2}