Index laws and surds: an index ladder, an index line for the three laws, and a solver that matches bases
A square root is just a power of one half
Every surd is a power in disguise: √2 is 2 to the power of ½, and ⅛ is 2 to the power of −3. One small set of index laws runs whole numbers, zero, negatives, fractions and surds alike. Change the index and watch each step up the ladder multiply by the base.
Index laws and surds: the ladderIndex 3/2
Change the index. Watch which way you step and what each step does.
…
Base
Index steps
Index3/2
Exact value
2√2
Decimal
2.828
Three laws on one lineMultiply
Pick a law, then change each index. Follow the jumps along the line.
…
Law
First index m1/2
Second index n3/2
Index of answer
2
Value
4
Solve by matching basesx = 3/4
Pick the left side, then change the right side. Watch where x lands.
…
Left side
Right side, as a power of 22^3/2
Solution x
3/4
Right side
2√2
Know this
Multiplying powers of one base is adding the indices, so a negative index means a reciprocal and a fractional index means a root: 2−1 = ½ and 21/2 = √2.