Exponents and radicals
Exponent questions look varied and are not. Almost every one reduces to five rules and one translation between radicals and powers. The variation is decoration, and the traps are consistent enough to name in advance.
The five rules
Learn these as moves, not as a list to recite. Each one changes the shape of an expression without changing its value, which is exactly what an equivalent-expression question is asking for.
- Multiplying powers: aᵐ · aⁿ = aᵐ⁺ⁿ. Add the exponents
- Dividing powers: aᵐ / aⁿ = aᵐ⁻ⁿ. Subtract the exponents
- Power of a power: (aᵐ)ⁿ = aᵐⁿ. Multiply the exponents
- Zero and negative: a⁰ = 1, and a⁻ⁿ = 1 / aⁿ
- Power of a product: (ab)ⁿ = aⁿbⁿ, and (a/b)ⁿ = aⁿ / bⁿ
A radical is a fractional power
The square root of a is a to the power one half, and the nth root of a to the m is a to the power m over n. Writing it as a power is what makes most compression questions collapse: the fourth root of x cubed times the square root of x becomes x to the three quarters times x to one half, so the exponents add to five quarters. The radical sign hides that addition, and the power makes it visible.
Simplifying a radical
Pull out the largest perfect square you can find, then leave the rest under the sign. The square root of 50 is the square root of 25 times 2, which is 5 root 2. The square root of 72 is the square root of 36 times 2, which is 6 root 2. You can add radicals only when the number under the sign matches, so root 2 plus root 8 becomes root 2 plus 2 root 2, which is 3 root 2.
The three traps
These are the errors the answer choices are built around, and each one is a rule applied to the wrong object.
- The root of a sum is not the sum of the roots: root of (9 + 16) is 5, not 7
- A coefficient on the outside is not a base: 2x² cubed is 8x⁶, not 2x⁶
- A negative sign is not part of the base unless it is inside brackets: -3² is -9, while (-3)² is 9
Working backwards from a value
When an equation gives you a result such as 16 to the three quarters, do not reach for the calculator. Take the fourth root first, which is 2, then cube it, which is 8. Root first, power second, because the root of a small number stays small and the arithmetic stays in your head.
What does x to the power one half mean?
It is the square root of x, for x at least zero. In general, a to the power m over n is the nth root of a to the m, so the denominator is the root and the numerator is the power.
How do I simplify the square root of 72?
Find the largest perfect square factor: 72 is 36 times 2, so the root is 6 root 2.
Can I add two radicals together?
Only when the number under the sign is the same. Root 2 plus root 8 becomes root 2 plus 2 root 2, which is 3 root 2. Root 2 plus root 3 cannot be combined.
Why is a negative exponent a fraction?
Because dividing by a power and multiplying by its reciprocal do the same thing. The negative sign moves the power across the fraction bar, so a to the minus n is 1 over a to the n.
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