Grade by Grade
B. Exponential Functions · Exponential Functions

Rational Exponents, Part 1: What Does 9^(1/2) Mean?

B1.1 B1.2About 70 minutes
Learning goals
  • I can use the exponent rules to explain why x^(1/2) = √x and x^(1/3) = ∛x.
  • I can evaluate powers of the form x^(1/n) (x > 0) without a calculator when x is a perfect power, and with a calculator otherwise.
  • I can evaluate powers with negative unit-fraction exponents, such as 64^(−1/3).
  • I can enter fraction exponents correctly on a calculator using brackets.

Words to know

rational exponent
An exponent that is a fraction (a rational number), such as 1/2, 2/3 or −3/4.
square root
√x is the positive number that multiplied by itself gives x. √49 = 7.
cube root
∛x is the number that used as a factor three times gives x. ∛125 = 5, because 5 × 5 × 5 = 125.
nth root
ⁿ√x is the number that used as a factor n times gives x. ⁴√81 = 3 because 3⁴ = 81.
x^(1/n)
Another way to write the nth root: x^(1/n) = ⁿ√x, for x > 0.
perfect power
A number that is an exact power of a whole number, such as 64 = 2⁶ = 4³ = 8².

A power halfway between two powers

In a simple model, a colony of rabbits doubles every year: 1 rabbit pair now, 2 after one year, 4 after two years. How many after half a year? Not 1.5. Doubling happens smoothly, so after half a year the population has been multiplied by some number m, and after another half year by m again. Two half-years make one doubling, so m × m = 2. That means m = √2 ≈ 1.414.

In exponent language, 2^(1/2) × 2^(1/2) should equal 2¹ = 2. So 2^(1/2) = √2. Fraction exponents are not strange at all: they are what you need for any time that is not a whole number.

How we write fraction exponents here

A small raised fraction is hard to type, so on this site we write "9 to the power 1/2" as 9^(1/2). The brackets show the whole fraction is the exponent. This is also exactly how you type it on a calculator: 9 ^ ( 1 ÷ 2 ). Whole-number exponents are written the usual way, as in 9².

Investigating x^(1/2)

We want the exponent rules to keep working. Use the product rule on 9^(1/2) × 9^(1/2):

9^(1/2) × 9^(1/2) = 9^(1/2 + 1/2) = 9¹ = 9

So 9^(1/2) is a number that, multiplied by itself, gives 9. That is the definition of the square root: 9^(1/2) = √9 = 3.

Investigating x^(1/3)

Three equal factors: 8^(1/3) × 8^(1/3) × 8^(1/3) = 8^(1/3 + 1/3 + 1/3) = 8¹ = 8.

So 8^(1/3) is the number that used three times as a factor gives 8. That is the cube root: 8^(1/3) = ∛8 = 2.

The general rule

For x > 0 and a positive integer n:

x^(1/n) = ⁿ√x

The power of a power rule agrees: (x^(1/n))ⁿ = x^(n/n) = x¹ = x. The denominator of the exponent is the index of the root.

side = 3area = 9 square units9^(1/2) = √9 = 3edge = 2volume = 8 cubic units8^(1/3) = ∛8 = 2
A square with area 9 has side 9^(1/2) = 3. A cube built from 8 small cubes has edge 8^(1/3) = 2. Square roots undo squaring; cube roots undo cubing.
PowerMeansValueCheck
16^(1/2)√1644² = 16
16^(1/4)⁴√1622⁴ = 16
27^(1/3)∛2733³ = 27
32^(1/5)⁵√3222⁵ = 32
1000^(1/3)∛10001010³ = 1000
(1/9)^(1/2)√(1/9)1/3(1/3)² = 1/9
To check a root, raise your answer to the index. You should get the base back.

Seeing it on the graph

If you plot y = 2ˣ for whole-number x you get the points (0, 1), (1, 2), (2, 4), (3, 8). Joining them with a smooth curve says there are values in between. The curve at x = 1/2 has height √2 ≈ 1.41, and at x = 3/2 it has height 2^(3/2) = (√2)³ ≈ 2.83. Fraction exponents fill in the gaps of the graph.

Why the base must be positive here

In this course, a base with a fraction exponent is always positive (x > 0). Try (−4)^(1/2) on a calculator: you get an error, because no real number times itself gives −4. Some cases, such as ∛(−8) = −2, do work, but to keep the rules reliable we only use positive bases with rational exponents.

xy−1123123456780(1/2, 1.41)(3/2, 2.83)y = 2ˣ
The smooth curve y = 2ˣ passes through (1/2, √2) and (3/2, 2^(3/2)). Fraction exponents give the heights of the curve between the whole-number points.
Worked example

Example 1: Evaluate without a calculator

Evaluate (a) 49^(1/2) (b) 125^(1/3) (c) 81^(1/4) (d) (16/81)^(1/4) (e) 0.001^(1/3)

  1. (a) √49 = 7 (7² = 49).
  2. (b) ∛125 = 5 (5³ = 125).
  3. (c) ⁴√81 = 3 (3⁴ = 81).
  4. (d) Take the root of the top and the bottom: ⁴√16 / ⁴√81 = 2/3. Check: (2/3)⁴ = 16/81. ✓
  5. (e) 0.001 = 1/1000, and ∛(1/1000) = 1/10 = 0.1. Check: 0.1³ = 0.001. ✓

7, 5, 3, 2/3, 0.1

Worked example

Example 2: A negative fraction exponent

Evaluate (a) 64^(−1/3) (b) (1/32)^(−1/5)

  1. A negative exponent still means "reciprocal". The fraction part means "root". Do them in either order.
  2. (a) 64^(−1/3) = 1 / 64^(1/3) = 1 / ∛64 = 1/4.
  3. (b) Flip the base first: (1/32)^(−1/5) = 32^(1/5) = ⁵√32 = 2.

1/4 and 2

Worked example

Example 3: Use a calculator correctly

Evaluate 20^(1/3) to three decimal places. Then explain what goes wrong if you type 20 ^ 1 ÷ 3.

  1. Type 20 ^ ( 1 ÷ 3 ) =. The display shows 2.714 417 6…, so 20^(1/3) ≈ 2.714.
  2. Check: 2.714³ ≈ 19.99, close to 20. ✓ (It makes sense: ∛8 = 2 and ∛27 = 3, and 20 is between 8 and 27.)
  3. Without brackets the calculator does the power first: 20¹ = 20, then ÷ 3 gives 6.667. That is (20¹) ÷ 3, not 20^(1/3).

20^(1/3) ≈ 2.714. Always put a fraction exponent in brackets.

Just for fun

Why is √2 so hard to have a conversation with?

It never knows when to stop: √2 = 1.414 213 56… and its decimals go on forever without repeating.

Hands-on activity

Build roots with cubes and coins

You need: 27 small cubes (dice, sugar cubes or building bricks all the same size work well), and about 25 coins.

  1. Square roots: arrange 16 coins in a square. How many coins are along one side? Do the same with 25 coins. Record: 16^(1/2) = ?, 25^(1/2) = ?
  2. Try 20 coins. Can you make a square? What does that say about 20^(1/2)?
  3. Cube roots: stack 8 cubes into one big cube. How many cubes along each edge? Now try 27 cubes.
  4. Use a calculator to find 20^(1/2) and compare with what you saw in step 2.

What to look for: 16^(1/2) = 4 and 25^(1/2) = 5. Twenty coins cannot make a square, because 20^(1/2) ≈ 4.47 is not a whole number (it is between 4 and 5). Eight cubes make a 2 × 2 × 2 cube, so 8^(1/3) = 2, and 27 cubes make a 3 × 3 × 3 cube, so 27^(1/3) = 3.

Your turn

1

Evaluate: 36^(1/2)

2

Evaluate: 216^(1/3)

3

Evaluate: 625^(1/4)

4

Evaluate: (25/49)^(1/2)

5

Evaluate: 100^(−1/2)

6

Evaluate: (1/8)^(−1/3)

7

Use a calculator to evaluate 50^(1/2) and 50^(1/3) to two decimal places.

8

Write 7^(1/5) using a root sign.

More practice: warm-up

1

Evaluate: 4^(1/2)

2

Evaluate: 27^(1/3)

3

Evaluate: 1^(1/7)

4

Evaluate: 10 000^(1/4)

5

Write √11 as a power with a fraction exponent.

More practice: on your own

1

Evaluate: 0.04^(1/2)

2

Evaluate: (27/1000)^(1/3)

3

Evaluate: 81^(−1/4)

4

Between which two whole numbers is 40^(1/3)? Explain without a calculator, then check.

5

Sam typed 64 ^ 1 ÷ 2 and got 32. What did Sam actually calculate, and what is 64^(1/2)?

Challenge

1

Show that 64^(1/2), 64^(1/3) and 64^(1/6) are all whole numbers, and find them.

2

A cube-shaped water tank holds 3.375 m³. How long is each edge?

3

The rabbit colony in the opening model doubles every year. Use 2^(1/4) to find the factor by which it grows every 3 months (to 3 decimal places). Check that four of these growth steps give a doubling.

Check your understanding

Quiz

1.

What does 25^(1/2) equal?

2.

Which expression equals ∛x?

3.

Evaluate 16^(−1/4).

4.

Why does x^(1/2) × x^(1/2) = x tell us that x^(1/2) = √x?

5.

Evaluate (8/27)^(1/3).

6.

Which calculator entry gives 20^(1/3)?

For parents and teachers

Big idea (B1.1, B1.2): students determine, through investigation, what a fraction exponent means. A power like 9^(1/2) is the square root of 9, and x^(1/n) is the nth root of x. The key reasoning is that the exponent rules must keep working: 9^(1/2) × 9^(1/2) = 9¹.

How to help: Ask your teen to check each root by raising the answer back to the power ("Is 5⁴ really 625?"). Estimating first ("∛40 is between 3 and 4") catches most calculator mistakes.

Common difficulties:

  • Thinking x^(1/2) means x ÷ 2: 16^(1/2) is 4, not 8.
  • Calculator brackets: 20 ^ 1 ÷ 3 is not 20^(1/3).
  • Negative exponents: 64^(−1/3) is 1/4, not −4.
  • Negative bases: in this course rational exponents are only used with positive bases. (−4)^(1/2) is not a real number.