Fractional Exponents: The 1/2 Rule
Discover that a^(1/2) is just another way to write the square root of a.
Equation
y = x^1
Graph
Table
| x | y |
|---|---|
| 0 | 0 |
| 4 | 4 |
| 8 | 8 |
| 12 | 12 |
| 16 | 16 |
| 20 | 20 |
| 24 | 24 |
| 28 | 28 |
| 32 | 32 |
| 36 | 36 |
| 40 | 40 |
| 44 | 44 |
| 48 | 48 |
| 52 | 52 |
| 56 | 56 |
| 60 | 60 |
| 64 | 64 |
What this lesson covers
What you do
You shape the function y = x^e and watch the graph answer.
Challenges to clear
- Slide the exponent e to 0.5. Find the input x in the table where y = 8 — because 64^(1/2) = √64 = 8.
- Same rule elsewhere: √36 = 6 — check the x = 36 row.
- A power of one half IS a square root. Slide e to 0.5.
- Now the multiplier: x = 16 must read 12. √16 is 4, so a is 3.
Check yourself
Why does the expression x^(1/2) give the same result as the square root symbol √x?
If y = x^(1/2) represents the side length of a square with area x, what does the exponent 1/2 represent geometrically?
- Because raising a number to the power of 1/2 is the inverse operation of squaring it. — correct
- Because 1/2 is half of 1, so it divides the number by 2.
- Because the exponent 1/2 means you multiply the base by 0.5.
- Because square roots only work for perfect squares.
- It represents finding the side length from the area (undoing the squaring process). — correct
- It represents doubling the area of the square.
- It represents calculating the perimeter of the square.
- It represents halving the side length of the square.