Stretch exponents from whole numbers to fractions — and a root becomes a power: √a = a1/2, am/n = n√(am).
You already know that an means "multiply a by itself n times," that a1 = a, and that a negative exponent flips a power over: a−n = 1/an. But what could a half power possibly mean — what is 21/2, or 82/3? This lesson finishes the story of exponents. It ties a root to a fractional power, shows that am/n is simply "take the n-th root, then the m-th power," and proves that the same five power rules still hold for every fraction. Once they do, we can finally put a number on 20.5, 21.7, and every exponent in between — and collect the dots that will draw a smooth exponential curve next lesson.
Start where the old story ended. For a whole number n, the power an is repeated multiplication: an = a·a·…·a, with n factors. So 23 = 2·2·2 = 8, and 21 is just a single factor, 2. Two special values complete the integer picture — and they are defined so the pattern never breaks.
First, a0 = 1 for any a ≠ 0. Why? Going up the ladder of powers of 2, each step multiplies by 2: 21 = 2, 22 = 4, 23 = 8. Going down the ladder, each step must divide by 2. From 21 = 2 one more step down lands on 20 = 2 ÷ 2 = 1. Keep going and the pattern forces the negative exponents: a−n = 1/an, so 2−1 = ½ and 2−2 = ¼. A negative exponent does not make a number negative — it flips the power over into a fraction.
| power | 23 | 22 | 21 | 20 | 2−1 | 2−2 |
|---|---|---|---|---|---|---|
| value | 8 | 4 | 2 | 1 | ½ | ¼ |
The integer exponents already obey one clean rhythm: each step right on the exponent multiplies the value by the base, each step left divides by it. We met this in Powers and Exponents (Stage 6). The whole point of this lesson is to keep that rhythm going between the integers.
A square root asks a backward question. Instead of "what is 3 squared?" it asks "what number, squared, gives this?" Since 32 = 9, we write √9 = 3. The square root undoes the square.
Generalize the question to any power. The n-th root n√a asks: "what number, raised to the n-th power, gives a?" So
One care is needed about signs. For an even root (square, fourth, …), the number under the radical must be ≥ 0, and the symbol returns the principal (non-negative) root: √9 = 3, not −3, even though (−3)2 = 9 too. For an odd root (cube, fifth, …) there is no such worry — every real number has exactly one real odd root, and it can be negative: ∛(−8) = −2, since (−2)3 = −8.
A square root and a square don't quite cancel for negatives: √(a2) = |a|, not a. Check it: √((−3)2) = √9 = 3 = |−3|. The principal-root rule keeps the answer non-negative, so the absolute value is no accident.
Here is the bridge that completes the power operations. We want a single power, written a1/2, that behaves like √a. What must it satisfy? A square root, squared, gives the number back: (√a)2 = a. So whatever a1/2 is, it should obey (a1/2)2 = a. And by the old power rule (am)n = amn,
(a1/2)2 = a(1/2)·2 = a1 = a. ✓
The exponent rule does the work for us: squaring a "half power" doubles the exponent from ½ to 1. So a1/2 squares to a — it is the square root. We define
√a = a1/2
The same reasoning works for any root. Raising a1/n to the n-th power multiplies its exponent by n: (a1/n)n = a(1/n)·n = a1 = a. So a1/n is the number whose n-th power is a — exactly the n-th root:
n√a = a1/n
Check it against numbers we already know: 91/2 = √9 = 3; 81/3 = ∛8 = 2; 161/4 = 4√16 = 2. Taking a root is nothing more exotic than raising to a unit fraction power.
Pick a base and an index n. Watch n√a become a1/n — the same number, two notations.
A root is a unit-fraction power. The denominator of the exponent is the index of the root: n√a = a1/n. So a1/2 is not "a ÷ 2" — it is the square root of a. (Quick gut-check: 91/2 = 3, while 9 ÷ 2 = 4.5. Different answers.)
Unit fractions were the warm-up. What about am/n with a numerator other than 1, such as 82/3? Split the exponent into its two jobs using the very same power rules. Since mn = m · 1n = 1n · m, we can write
am/n = (a1/n)m = (n√a)m and am/n = (am)1/n = n√(am).
In words: am/n is the n-th root and the m-th power, in either order. The denominator n is the root; the numerator m is the power. Doing the root first usually keeps the numbers small, so reach for that order.
82/3 = (∛8)2 = 22 = 4. Check the other order: ∛(82) = ∛64 = 4. Same answer. ✓
43/2 = (√4)3 = 23 = 8.
163/4 = (4√16)3 = 23 = 8.
9−1/2 = 1/91/2 = 1/√9 = ⅓. (The minus flips it; the half takes the root.)
Set the power m and the root n over base a = 8. Both orders are computed — confirm they agree.
A negative base with a fractional exponent is treacherous: (−8)1/2 asks for a real square root of a negative number, and there is none. To keep every fractional power clean and to draw a smooth curve later, we take the base a > 0 throughout this stage.
The payoff of defining fractional exponents this way is that nothing new has to be learned. The five power rules you already know now cover all rational exponents — integers, unit fractions, and everything in between behave alike.
| Rule | Statement | Fractional example |
|---|---|---|
| product | am·an = am+n | a1/2·a1/3 = a5/6 |
| quotient | am/an = am−n | a3/4/a1/4 = a1/2 |
| power of a power | (am)n = amn | (a1/3)3 = a1 = a |
| power of a product | (ab)n = anbn | (ab)1/2 = √a·√b |
| power of a quotient | (a/b)n = an/bn | (a/b)1/2 = √a/√b |
Each example is worth checking against meaning, not memory. The product rule says that adding exponents multiplies powers — so multiplying two square roots of a should add ½ + ½ = 1: indeed a1/2·a1/2 = a1 = a, i.e. √a·√a = a. The power-of-a-power rule says (a1/3)3 = a(1/3)·3 = a — cubing a cube root returns the number, exactly the n-th-root definition again. And mixing denominators is fine: a1/2·a1/3 = a1/2 + 1/3 = a5/6.
The product rule adds exponents — it does not multiply them: am·an = am+n, never amn. Test it: 22·23 = 4·8 = 32 = 25 = 22+3. If you multiplied the exponents you would get 26 = 64, which is wrong.
With fractional exponents defined and the rules intact, every rational exponent now has a value. Some are exact, like 82/3 = 4; most are irrational and we reach them with a calculator:
20.5 ≈ 1.414, 21.5 ≈ 2.828, 21.7 ≈ 3.249, 2−1 = 0.5.
Notice that a bigger exponent gives a bigger value, because the base 2 is larger than 1 — each tick to the right multiplies by a bit more of 2. (Compare 21.5 ≈ 2.828 with 21.7 ≈ 3.249: more exponent, more value.) That single fact will decide which of two powers is larger throughout this stage.
Now let the exponent x run through a grid of values and tabulate y = 2x. Plot each pair (x, 2x) as a point. The dots no longer leave gaps — they line up along a rising, smoothly bending shape:
Step x through the grid. Each value drops an amber dot at (x, 2x); together they want to join into one curve.
Because fractional and irrational exponents now have values, the dots no longer skip — they want to join into one continuous, unbroken curve: the exponential function y = 2x. Stretching exponents from whole numbers down to fractions is exactly what makes that smooth curve possible.
The power operations are now complete down to every fraction:
Write √7 as a power of 7.
A square root is a unit-fraction power: √7 = 71/2.
Evaluate 271/3.
271/3 = ∛27 = 3, because 33 = 27.
Evaluate 163/4.
Root first, then power: 163/4 = (4√16)3 = 23 = 8.
Evaluate 25−1/2.
The minus flips, the half takes the square root: 25−1/2 = 1/251/2 = 1/√25 = ⅕.
Simplify a1/2·a1/2 (with a > 0).
Add the exponents: a1/2·a1/2 = a1/2 + 1/2 = a1 = a. (That is just √a·√a = a.)
Which is larger, 21.5 or 21.7? Explain why without a calculator.
21.7 is larger. The base 2 is greater than 1, so a bigger exponent gives a bigger value — and 1.7 > 1.5. (Numerically 21.5 ≈ 2.828 < 3.249 ≈ 21.7.)
Six questions to lock it in. Tap the answer you think is right.
The big idea is one of continuity of meaning: we already accepted a0 = 1 and a−n = 1/an because they kept the power rules unbroken; fractional exponents are defined the same way. We insist that a1/n obey (a1/n)n = a, which forces a1/n = n√a — so a root is a power, and the whole machinery carries over to rational exponents at no extra cost. From there am/n = n√(am) = (n√a)m follows directly.
Three misconceptions are worth heading off. First, students read a1/2 as "a divided by 2"; the cure is the gut-check 91/2 = 3 versus 9 ÷ 2 = 4.5. Second, they multiply exponents when multiplying powers — aman "=" amn — instead of adding; a tiny numerical test (22·23 = 32 = 25) settles it. Third, they forget that even roots require a non-negative radicand and return the principal value, so √(a2) = |a|, not a. Keeping the base positive throughout sidesteps the thorniest sign issues and prepares the smooth curve of the next lesson.
Common Core: N-RN.A.1 (explain how the definition of rational exponents follows from extending the properties of integer exponents) and N-RN.A.2 (rewrite expressions involving radicals and rational exponents). This lesson opens Stage 24; it feeds directly into graphing exponential functions (F-IF.C.7e) next.