Grade by Grade
B. Exponential Functions · Exponential Functions

Rational Exponents, Part 2: Powers Like 8^(2/3) and 16^(0.75)

B1.1 B1.2About 70 minutes
Learning goals
  • I can explain why x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ) for x > 0, using the power of a power rule.
  • I can evaluate powers with rational exponents, including negative ones, without a calculator when the base is a perfect power.
  • I can change a decimal exponent to a fraction (0.75 = 3/4) to evaluate a power.
  • I can evaluate any power with a rational exponent with a calculator and judge if the answer is reasonable.

Words to know

x^(m/n)
For x > 0: the nth root of x, raised to the power m. x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ).
numerator of the exponent
In x^(m/n), m: the power.
denominator of the exponent
In x^(m/n), n: the root (the index).
index
The small number on a root sign that tells which root: ∛ has index 3.
radical
An expression with a root sign, such as √5 or ∛20.

How big is the face of the box?

A cube-shaped storage box has a volume of 125 L, and 1 L = 1000 cm³, so the volume is 125 000 cm³. How much cardboard does one face need? The edge is the cube root of the volume, and the face area is the edge squared. So the face area is (∛V)², which you can also write as V^(2/3). For this box: (∛125 000)² = 50² = 2500 cm².

That 2/3 exponent does two jobs at once: the 3 takes a cube root and the 2 squares. This lesson shows how every fraction exponent works the same way.

Splitting a fraction exponent

The exponent 2/3 is 2 × 1/3 (or 1/3 × 2). The power of a power rule says (aᵐ)ⁿ = aᵐⁿ, so:

  • 8^(2/3) = (8^(1/3))² = (∛8)² = 2² = 4 (root, then power)
  • 8^(2/3) = (8²)^(1/3) = ∛64 = 4 (power, then root)

In general, for x > 0 and integers m and n with n > 0:

x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ)

Memory aid: the denominator is the root (it is "down" like a root in the ground) and the numerator is the power.

8^(2/3)root first(∛8)² = 2²power first∛(8²) = ∛644Both roads arrive at 4. Taking the root first keeps the numbers small.
Two ways to evaluate 8^(2/3). Both give 4, but taking the root first means you never have to find the cube root of a big number.

Step by step on a number line

Count up from 8⁰ = 1 in steps of 1/3 in the exponent. Each step of 1/3 multiplies by 8^(1/3) = 2:

8⁰ = 1, 8^(1/3) = 2, 8^(2/3) = 4, 8^(3/3) = 8¹ = 8.

Equal steps in the exponent always mean equal multiplying factors, never equal added amounts. That is the signature of exponential growth, which you will use for the rest of this unit.

012345678×2×2×2
8⁰ = 1, 8^(1/3) = 2, 8^(2/3) = 4 and 8¹ = 8. Each step of 1/3 in the exponent multiplies by ∛8 = 2.
Worked example

Example 1: Evaluate without a calculator

Evaluate (a) 27^(2/3) (b) 16^(3/4) (c) 4^(5/2) (d) (9/25)^(3/2)

  1. (a) Root first: ∛27 = 3, then 3² = 9.
  2. (b) ⁴√16 = 2, then 2³ = 8.
  3. (c) √4 = 2, then 2⁵ = 32.
  4. (d) √(9/25) = 3/5, then (3/5)³ = 27/125.

9, 8, 32, 27/125

Worked example

Example 2: Negative rational exponents

Evaluate (a) 27^(−2/3) (b) (16/81)^(−3/4)

  1. Three jobs: the minus sign means reciprocal, the denominator means root, the numerator means power.
  2. (a) 27^(−2/3) = 1 / 27^(2/3) = 1 / (∛27)² = 1 / 3² = 1/9.
  3. (b) Flip the base to remove the minus sign: (81/16)^(3/4). Then ⁴√(81/16) = 3/2, and (3/2)³ = 27/8.

1/9 and 27/8 (3.375)

Worked example

Example 3: Decimal exponents

Evaluate (a) 16^0.75 (b) 32^0.4 (c) 100^1.5

  1. Change each decimal exponent to a fraction in lowest terms.
  2. (a) 0.75 = 3/4. ⁴√16 = 2 and 2³ = 8.
  3. (b) 0.4 = 2/5. ⁵√32 = 2 and 2² = 4.
  4. (c) 1.5 = 3/2. √100 = 10 and 10³ = 1000.

8, 4, 1000

Worked example

Example 4: When the root is not a whole number

Evaluate 5^(2/3) to three decimal places and check that it is reasonable.

  1. Calculator: 5 ^ ( 2 ÷ 3 ) = 2.924 017… ≈ 2.924.
  2. Reasonable? The exponent 2/3 is between 0 and 1, so the answer should be between 5⁰ = 1 and 5¹ = 5. ✓
  3. Another check: 2.924³ ≈ 25.0, and 5² = 25. ✓ (Cubing x^(2/3) gives x².)

5^(2/3) ≈ 2.924

Three jobs, any order

For x^(−m/n):

  • minus sign → reciprocal
  • n (denominator) → nth root
  • m (numerator) → mth power

You may do them in any order, but "flip, root, power" usually keeps the numbers smallest. Always check: a positive exponent less than 1 on a base greater than 1 gives an answer between 1 and the base.

Just for fun

Why did 8 feel a little down after being raised to the power 1/3?

It had been cut down to size: it was only 2.

Hands-on activity

From volume to face area with a real box

You need: a cube-shaped (or nearly cube-shaped) box or container, a ruler or measuring tape, and a calculator.

  1. Measure the three edges in centimetres. If they differ a little, use their average as the edge length e.
  2. Calculate the volume V = e³.
  3. Now pretend you only know V. Calculate V^(2/3) on your calculator: V ^ ( 2 ÷ 3 ).
  4. Calculate the area of one face directly: e².
  5. Compare steps 3 and 4. Then explain in one sentence why V^(2/3) gives the face area.

What to look for: steps 3 and 4 give the same number (up to rounding). For example, a 12 cm cube has V = 1728 cm³ and 1728^(2/3) = (∛1728)² = 12² = 144 cm², the area of one face. The cube root turns volume into edge length, and squaring turns edge length into area.

Your turn

1

Evaluate: 64^(2/3)

2

Evaluate: 9^(3/2)

3

Evaluate: 32^(3/5)

4

Evaluate: 125^(−2/3)

5

Evaluate: (8/27)^(2/3)

6

Evaluate: 81^0.75

7

Write 7^(3/4) in radical form two ways.

8

Evaluate 10^(3/2) with a calculator to two decimal places.

More practice: warm-up

1

In 27^(2/3), which number tells you the root?

2

Evaluate: 4^(3/2)

3

Evaluate: 1000^(2/3)

4

Write 0.5 as a fraction and evaluate 49^0.5.

5

Evaluate: 8^(4/3)

More practice: on your own

1

Evaluate: (1/16)^(−3/4)

2

Evaluate: 0.01^(3/2)

3

Evaluate: 243^0.6

4

Without a calculator, decide whether 20^(3/4) is between 1 and 20, or greater than 20. Then evaluate it to one decimal place.

5

A cube has volume 343 cm³. Use a rational exponent to find the area of one face.

Challenge

1

Evaluate: (27^(−1/3) + 9^(1/2))^(−1)

2

Find x if x^(2/3) = 36 (x > 0).

3

Explain why 4^(2/4) and 4^(1/2) are equal, and evaluate both. Then evaluate 64^(4/6).

Check your understanding

Quiz

1.

Which expression equals 27^(2/3)?

2.

Evaluate 16^(3/2).

3.

Evaluate 8^(−2/3).

4.

What is 0.75 as a fraction exponent, and what is 16^0.75?

5.

Evaluate (4/9)^(3/2).

6.

Without a calculator, which is closest to 50^(1/2)?

For parents and teachers

Big idea (B1.1, B1.2): students extend fraction exponents to x^(m/n): the denominator is a root and the numerator is a power, and a negative sign means reciprocal. They evaluate such powers exactly for perfect powers and with a calculator otherwise.

How to help: Have your teen say the three jobs aloud for an exponent like −2/3: "reciprocal, cube root, square". Ask for an estimate before every calculator answer.

Common difficulties:

  • Mixing up root and power: 27^(2/3) is (∛27)² = 9, not (√27)³.
  • Multiplying base by exponent: 16^(3/2) is 64, not 24.
  • Decimal exponents: change them to fractions in lowest terms first (0.4 = 2/5).
  • Calculator brackets: type 5 ^ ( 2 ÷ 3 ), not 5 ^ 2 ÷ 3.