For example:
$ 2^3 = 2 \cdot 2 \cdot2 $
But what about a fractional power such as:
$$ 2^{2/3} = ? $$
1) How would I explain this?
2) How would I find value?
3) How would computers calculate this number?
Asked
Active
Viewed 50 times
0
-
- The first line is not a definition, rather the result of the definition for integers. 2. By plugging in the value of the definition of an exponential function. 3. As far as I know they use power series approximation due to the fact that we know the accuracy of number of decimal places by the number of terms it calculates.
– Matthew Liu Feb 08 '19 at 16:07 -
@MatthewLiu Can you elaborate question 3? – Jundullah Feb 08 '19 at 16:35
2 Answers
0
Hint: $$2\times 2\times 2=2^{3}$$ and $$(2^{3})^{1/3}=2$$ and $$2^{2/3}=\sqrt[3]{4}=\sqrt[3]{\frac{8}{2}}=\frac{2}{\sqrt[3]{2}}$$
Dr. Sonnhard Graubner
- 95,283
-
-
A computer with a Computer Algebra System calculates this number as i have written. – Dr. Sonnhard Graubner Feb 08 '19 at 16:57
0
For question #3: In the past, when computers didn't have the vast amounts of memory they do now, finding a root often fell to such methods as bisection, Newton's method, regula falsi, the secant method, and others. Plus, you had to write the programs, put them on cards, and be restricted to a few decimal places. These days, there are programs that will calculate your root to whatever accuracy you wish - I don't know what algorithm they use, but often it's one that reduces the amount of time and the amount of error in finding the root.
bjcolby15
- 3,599