Properties of Powers and Roots

Welcome visitors. At this meeting the material will be presented with examples of questions that are easy to understand.

Here is a formula that you should remember:
1. if $\displaystyle a^b=a^c$ then $b=c$
2. $\displaystyle a^b.a^c=a^{b+c}$
3. $\displaystyle \frac{a^b}{a^c}=a^{b-c}$
4. $\displaystyle \left(a^b \right)^c=a^{b.c}$
5. $\displaystyle a^{-b}=\frac{1}{a^b}$
6. $\displaystyle a^b.c^b=(a.c)^b$
7. $\displaystyle \frac{a^b}{c^b}=\left(\frac{a}{c}\right)^b$
8. $\displaystyle a^{\frac{b}{c}}=\sqrt[c]{a^b}$

Then the following will be given examples that match the 8 formulas or properties above.

1. The $k$ value of the $\displaystyle 2^{3k+1}=4^{k-5}$ equation is …
Solution: $$2^{3k+1}=\left(2^2\right)^{k-5}$$ $$2^{3k+1}=2^{2k-10}$$ $$3k+1=2k-10$$ $$3k-2k=-10-1$$ $$k=-11$$
2. The $m$ value of the $\displaystyle 5^{10m-7}.5^m=25$ equation is …
Solution: $$5^{10m-7}.5^m=5^{10m-7+m}$$ $$=5^{11m-7}=5^2$$ $$11m-7=2$$ $$m=\frac{9}{11}$$
3. Prove that $x^0=1$ for any number $x$
Solution: $$x^0=x^{a-a}$$ $$x^0=\frac{x^a}{x^a}=1$$ If in ordinary fractions it happens that the numerator and denominator are the same then the result is 1.

4. Count $\displaystyle \left(2^3 \right)^2$
Solution: $$=2^{3.(2)}=2^6=64$$
5. The solution of $\displaystyle \frac{1}{3^{x+1}}=27$ is …
Solution: $$\to 3^{-(x+1)}=3^3$$ $$-(x+1)=3$$ $$-x-1=3$$ $$x=-4$$
6. The solution of $\displaystyle 54^{x}.3^{-2x}=36$ is …
Solution: $$54^{x}.3^{-2x}=54^{x}.\left(3^{-2}\right)^x$$ $$=\left(54.\left(\frac{1}{9}\right)\right)^x=6^x$$ $$\to 6^x=6^2$$ $$x=2$$
7. The solution of $\displaystyle \frac{7^{b+1}}{8^{2b-3}}=7.(8^3)$ is …
Solution: $$\frac{7^{b+1}}{8^{2b-3}}=\frac{7.7^b}{64^b.8^{-3}}$$ $$7.(8^3).\left(\frac{7}{64}\right)^b=7.(8^3)$$ $$\left(\frac{7}{64}\right)^b=1$$ $$\left(\frac{7}{64}\right)^b=\left(\frac{7}{64}\right)^0$$ $$b=0$$

8. if $\displaystyle \sqrt[5]{10^{2d-3}}=100$ then $d=…$
Solution: $$\sqrt[5]{10^{2d-3}}=10^{\frac{2d-3}{5}}$$ $$10^{\frac{2d-3}{5}}=10^2$$ $$\frac{2d-3}{5}=2$$ $$2d-3=10$$ $$2d=10+3$$ $$d=\frac{13}{2}=6,5$$

Thus this post, see you in other posts and hopefully useful.