Cracking the Code: 8^(2/3)
Decode fractional exponents by breaking the fraction into a root and a power.
Equation
y = x^(2/1)
Graph
Table
| x | y |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 16 | 256 |
| 17 | 289 |
| 18 | 324 |
| 19 | 361 |
| 20 | 400 |
| 21 | 441 |
| 22 | 484 |
| 23 | 529 |
| 24 | 576 |
| 25 | 625 |
| 26 | 676 |
| 27 | 729 |
What this lesson covers
What you do
You shape the function y = x^(2/n) and watch the graph answer.
Challenges to clear
- At x=8, move n to 3. y should be 4. This proves 8^(2/3) = 4.
- Also 27^(2/3) = 9: cube root of 27 is 3, squared is 9. Check x = 27.
- Now BOTH numbers are yours. The root on the bottom is a fourth root — slide n to 4.
- Now the power on top: x = 16 must give 8. The fourth root of 16 is 2, and 2 cubed is 8, so m = 3.
Check yourself
Why does 8^(2/3) equal 4? Which step explains the calculation best?
If you wanted to calculate 27^(2/3), why is it easier to take the cube root first rather than squaring 27 first?
- The denominator 3 means take the cube root of 8 (which is 2), then the numerator 2 means square that result (2^2 = 4). — correct
- The numerator 2 means multiply 8 by 2 to get 16, then the denominator 3 means divide by 3.
- The fraction 2/3 means multiply 8 by 2/3 directly, which equals 16/3.
- The denominator 3 means cube 8 (8*8*8), then the numerator 2 means take the square root.
- Because the cube root of 27 is a small integer (3), making the final squaring step simple (3^2 = 9). — correct
- Because squaring 27 first makes the number too large for the calculator.
- Because the order of operations in algebra always demands roots before powers.
- Because 27 is a perfect square, so we should square it first.
Think about it
- 8^(2/3): the 3 below means cube root (8 → 2), the 2 above means square (2 → 4). What is 8^(2/3)?