B. Exponential Functions · Exponential Functions

Rational Exponents I: Powers with Exponent 1/n

B1.2About 65 minutes

Learning goals

I can use the exponent laws to explain why x^(1/2) = √x and x^(1/3) = ∛x.
I can evaluate powers of the form x^(1/n) and x^(−1/n) without a calculator.
I can switch between radical form and exponent form.
I can evaluate fractional powers correctly on a calculator.

Words to know

rational exponent
An exponent that is a fraction m/n, such as 1/2, 2/3 or −3/4.

square root
A number that, multiplied by itself, gives the original number. √25 = 5 because 5 × 5 = 25.

cube root
A number that, used as a factor three times, gives the original number. ∛64 = 4 because 4 × 4 × 4 = 64.

nth root
A number that, used as a factor n times, gives the original number. Written ⁿ√x.

radical
An expression with a root sign, such as √7 or ∛10. The number under the sign is the radicand.

principal root
The positive root of a positive number. √9 means +3, not −3.

What could 9 to the power one half possibly mean?

You know 9² = 9 × 9: two factors of 9. But how can you use half a factor of 9? Repeated multiplication cannot answer that. Instead, mathematicians asked a smarter question: what value should 9^(1/2) have so that the exponent laws still work? Once you answer that, fractional exponents stop being strange.

A quick word about writing: on a screen or calculator, an exponent that is a fraction is written with a caret: 9^(1/2) means 9 to the power 1/2. The brackets around the fraction matter.

Remember from Grade 9

The power of a power law: (aᵐ)ⁿ = aᵐⁿ. For example, (2³)² = 2⁶ = 64. The product law: aᵐ × aⁿ = aᵐ⁺ⁿ. You also used square roots in the Pythagorean theorem: √49 = 7.

Using the exponent laws to decide

Suppose 9^(1/2) is some number. Square it and use the power of a power law:

(9^(1/2))² = 9^((1/2) × 2) = 9¹ = 9

So 9^(1/2) is a number whose square is 9. That is the square root of 9: 9^(1/2) = √9 = 3.

You can also use the product law: 9^(1/2) × 9^(1/2) = 9^(1/2 + 1/2) = 9¹ = 9. Two equal factors that multiply to 9: each is 3.

The same reasoning works for thirds. (27^(1/3))³ = 27¹ = 27, so 27^(1/3) is the number whose cube is 27. That is the cube root: 27^(1/3) = ∛27 = 3.

The rule

For x > 0 and a positive integer n:

x^(1/n) = ⁿ√x, the positive number whose nth power is x.

x^(1/2) = √x (square root)
x^(1/3) = ∛x (cube root)
x^(1/4) = ∜x (fourth root), and so on.

In this course the base of a rational exponent is positive (x > 0). For example, (−4)^(1/2) would be the square root of −4, which is not a real number.

A square with area 9 has side length 9^(1/2) = 3. A cube with volume 27 has edge length 27^(1/3) = 3. Square roots undo squaring; cube roots undo cubing.

Seeing it on the graph

The graph of y = 4ˣ gives another view. Go halfway from x = 0 to x = 1. The y-value goes from 1 to 4, but not halfway (2.5): it is multiplied by the same factor twice. Two equal factors that multiply to 4 are 2 and 2. So at x = 0.5 the graph is at y = 2, and 4^(1/2) = 2 = √4. This is why the smooth curve of an exponential function makes sense between the integers.

On y = 4ˣ, the point halfway between x = 0 and x = 1 is (0.5, 2). Going from 1 to 2 to 4 multiplies by 2 each half-step, so 4^0.5 = 2.

Worked example

Example 1: Evaluating x^(1/n)

Evaluate (a) 49^(1/2) (b) 125^(1/3) (c) 81^(1/4) (d) 32^(1/5) (e) (1/16)^(1/4) (f) 0.001^(1/3)

(a) 49^(1/2) = √49 = 7, since 7² = 49.
(b) 125^(1/3) = ∛125 = 5, since 5³ = 125.
(c) 81^(1/4) = ∜81 = 3, since 3⁴ = 81.
(d) 32^(1/5) = ⁵√32 = 2, since 2⁵ = 32.
(e) Take the root of the top and the bottom: (1/16)^(1/4) = 1^(1/4) / 16^(1/4) = 1/2. Check: (1/2)⁴ = 1/16.
(f) 0.001 = 1/1000, so 0.001^(1/3) = 1/10 = 0.1. Check: 0.1³ = 0.001.

7, 5, 3, 2, 1/2 and 0.1.

Worked example

Example 2: Negative fractional exponents

Evaluate (a) 64^(−1/3) (b) 100^(−1/2) (c) (1/9)^(−1/2)

A negative exponent still means reciprocal: x^(−1/n) = 1 / x^(1/n).
(a) 64^(−1/3) = 1 / 64^(1/3) = 1 / 4 = 1/4.
(b) 100^(−1/2) = 1 / √100 = 1/10.
(c) Flip the fraction first: (1/9)^(−1/2) = 9^(1/2) = 3.

1/4, 1/10 and 3.

Worked example

Example 3: Using a calculator, and estimating

(a) Estimate 20^(1/2), then use a calculator. (b) Evaluate 10^(1/3) to three decimal places. (c) Rosa types 16 ^ 1 ÷ 2 and gets 8. What went wrong?

(a) 4² = 16 and 5² = 25, so √20 is between 4 and 5, a little closer to 4. Calculator: 20 ^ (1 ÷ 2) ≈ 4.472.
(b) Type 10 ^ ( 1 ÷ 3 ). Result: 2.154. Check: 2.154³ ≈ 9.99.
(c) Without brackets the calculator does 16¹ first and then divides by 2: 16 ÷ 2 = 8. The exponent must be in brackets: 16 ^ (1 ÷ 2) = 4.

√20 ≈ 4.472; 10^(1/3) ≈ 2.154; always put a fractional exponent in brackets.

Just for fun

A funny-but-true fact: the word radical comes from the Latin radix, meaning "root". So does the word radish. Every time you write a radical sign, you are, in a way, drawing a little vegetable.

Hands-on activity

Build roots with blocks

You need: about 30 small identical cubes (building bricks, dice or sugar cubes), 16 coins or square tiles, and a calculator.

Arrange 16 coins (or tiles) into a perfect square. How many are on each side? Write the result as a power with exponent 1/2.
Build the largest perfect cube you can with your cubes: try 8 cubes, then 27. How many cubes are along each edge? Write each as a power with exponent 1/3.
Try to make a perfect square from 20 coins. What happens? Use the calculator to find 20^(1/2).
Predict, then check on the calculator: 64^(1/2), 64^(1/3) and 64^(1/6).

What to look for: 16^(1/2) = 4 (a 4 by 4 square). 8^(1/3) = 2 and 27^(1/3) = 3 (2 by 2 by 2 and 3 by 3 by 3 cubes). 20 coins cannot make a perfect square, because 20^(1/2) ≈ 4.47 is not a whole number. 64^(1/2) = 8, 64^(1/3) = 4 and 64^(1/6) = 2, because 8² = 4³ = 2⁶ = 64.

Your turn

Evaluate 36^(1/2).

Evaluate 8^(1/3).

Evaluate 16^(1/4).

Evaluate (4/25)^(1/2).

Evaluate 27^(−1/3).

Evaluate 0.36^(1/2).

Use the power of a power law to explain why 125^(1/3) = 5.

Write 7^(1/5) in radical form, and write ∜11 as a power.

More practice: warm-up

Evaluate 100^(1/2).

Evaluate 1000^(1/3).

Evaluate 1^(1/7).

Evaluate 25^(−1/2).

Write √x as a power of x.

More practice: on your own

Evaluate 243^(1/5).

Evaluate (8/27)^(−1/3).

Evaluate 16^(1/2) + 16^(1/4) − 16^(−1/2).

Without a calculator, between which two whole numbers is 50^(1/3)? Then check with a calculator.

A square garden plot has an area of 0.81 m². Write its side length as a power and evaluate it.

Challenge

Show that 64^(1/6) = (64^(1/2))^(1/3) = (64^(1/3))^(1/2). What is the value?

Liam says 9^(1/2) = 4.5 because "half of 9 is 4.5". Explain his error using the exponent laws.

A cube-shaped storage box has a volume of 0.125 m³. Find its edge length, then find the area of one face.

Check your understanding

Quiz

What is 49^(1/2)?

What is 64^(1/3)?

Which is equal to 81^(−1/2)?

Which explains why x^(1/3) = ∛x?

What is (1/32)^(1/5)?

On a calculator, which entry gives 16^(1/2) correctly?

For parents and teachers

Big idea: Expectation B1.2 asks students to work out the value of powers with rational exponents through investigation: patterns, graphs and the exponent laws. This lesson covers exponents of the form 1/n; the next lesson covers m/n.

How to help: Ask "what number times itself gives 49?" and "what number used three times gives 64?" Memorizing the squares to 15² and the cubes to 5³ (and 10³) makes this unit much faster.

Common difficulties:

Dividing by the denominator: 9^(1/2) is 3, not 4.5. Have your teen check by squaring the answer.
Negative exponents: 25^(−1/2) is 1/5, a positive number.
Calculator entry without brackets around the fraction.
Writing √(a + b) = √a + √b. This is false: √(9 + 16) = 5, but √9 + √16 = 7.