B. Exponential Functions · Exponential Functions

Rational Exponents II: Powers with Exponent m/n

B1.2 B1.3About 70 minutes

Learning goals

I can explain why x^(m/n) = (ⁿ√x)ᵐ using the power of a power law.
I can evaluate powers with rational exponents, including negative exponents and fraction or decimal bases.
I can change decimal exponents to fractions to evaluate powers.
I can evaluate numeric expressions with several integer and rational powers.

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 m/n, the m: it tells the power.

denominator of the exponent
In m/n, the n: it tells the root (the index).

index
The small number on a radical sign that tells which root: the 3 in ∛x.

rational base
A base that is a fraction or a terminating or repeating decimal, such as 4/9 or 0.25.

Two jobs in one exponent

Last lesson, x^(1/n) meant the nth root. Now look at 8^(2/3). The exponent 2/3 is (1/3) × 2, so by the power of a power law:

8^(2/3) = (8^(1/3))² = 2² = 4

The exponent 2/3 does two jobs. The denominator 3 says "take the cube root". The numerator 2 says "square it". Since (1/3) × 2 = 2 × (1/3), you may also do the jobs in the other order: 8^(2/3) = (8²)^(1/3) = 64^(1/3) = 4. Same answer.

The rule

For x > 0 and integers m and n with n > 0:

x^(m/n) = (ⁿ√x)ᵐ = ⁿ√(xᵐ)

Denominator = root, numerator = power. ("Roots are underground": the root goes on the bottom.)
Taking the root first usually keeps the numbers small.
A negative exponent still means reciprocal: x^(−m/n) = 1 / x^(m/n).

Two routes to 8^(2/3). Cube root then square: 8 → 2 → 4. Square then cube root: 8 → 64 → 4. Both give 4, but the root-first route uses smaller numbers.

Checking with the graph of y = 8ˣ

On y = 8ˣ, going from x = 0 to x = 1 multiplies y by 8. Split that into three equal steps of 1/3. Each step must multiply by the same factor, and three equal factors that make 8 are 2 × 2 × 2. So the curve passes through (1/3, 2), (2/3, 4) and (1, 8). The graph agrees: 8^(2/3) = 4.

Each step of 1/3 to the right on y = 8ˣ multiplies y by 2, so 8^(1/3) = 2, 8^(2/3) = 4 and 8^(−1/3) = 1/2.

Worked example

Example 1: Evaluating x^(m/n)

Evaluate (a) 16^(3/4) (b) 27^(4/3) (c) 25^(3/2) (d) 32^(−3/5)

(a) Fourth root first: 16^(1/4) = 2. Then cube: 2³ = 8.
(b) Cube root first: 27^(1/3) = 3. Then fourth power: 3⁴ = 81.
(c) Square root first: √25 = 5. Then cube: 5³ = 125.
(d) 32^(1/5) = 2, and 2³ = 8. The exponent is negative, so take the reciprocal: 1/8.

8, 81, 125 and 1/8.

Worked example

Example 2: Rational bases

Evaluate (a) (4/9)^(3/2) (b) (27/8)^(−2/3) (c) 0.16^(3/2)

(a) Apply the exponent to the top and the bottom: (4/9)^(3/2) = 4^(3/2) / 9^(3/2) = 2³ / 3³ = 8/27.
(b) Negative exponent: flip the fraction first. (27/8)^(−2/3) = (8/27)^(2/3) = 2² / 3² = 4/9.
(c) Write 0.16 as 16/100. (16/100)^(3/2) = 4³ / 10³ = 64/1000 = 0.064.

8/27, 4/9 and 0.064.

Worked example

Example 3: Decimal exponents

Evaluate (a) 32^0.4 (b) 100^1.5 (c) 81^0.75

Change each decimal exponent to a fraction in lowest terms.
(a) 0.4 = 2/5. 32^(2/5) = (⁵√32)² = 2² = 4.
(b) 1.5 = 3/2. 100^(3/2) = (√100)³ = 10³ = 1000.
(c) 0.75 = 3/4. 81^(3/4) = (∜81)³ = 3³ = 27.

4, 1000 and 27.

Worked example

Example 4: Numeric expressions with several powers

Evaluate without a calculator: 8^(2/3) + (2/3)⁻² − 16^(−1/2) + 5⁰

8^(2/3) = 4.
(2/3)⁻² = (3/2)² = 9/4.
16^(−1/2) = 1/4.
5⁰ = 1.
Total: 4 + 9/4 − 1/4 + 1 = 5 + 8/4 = 5 + 2 = 7.

When the answer is not a whole number

Most rational powers are irrational numbers. Use a calculator with brackets: 5^(2/3) is typed 5 ^ ( 2 ÷ 3 ) ≈ 2.924, and 7^(−3/2) is 7 ^ ( − 3 ÷ 2 ) ≈ 0.054. A good habit is to estimate first: 5^(2/3) is between 1^(2/3) = 1 and 8^(2/3) = 4, so 2.924 is reasonable.

Just for fun

Teacher: "How did you find 27^(2/3) so fast?"

Student: "Easy. I got to the root of the problem, then squared things away." (The answer is 9.)

Hands-on activity

Rational-exponent matching game

You need: 16 sticky notes or small slips of paper, a pencil, and a family member to play against (or a timer).

On 8 slips write: 4^(3/2), 27^(2/3), 16^(−1/4), 1000^(2/3), (9/4)^(1/2), 32^(3/5), 81^(−3/4), 0.25^(−1/2).
On the other 8 slips write the values in a scrambled order: 8, 9, 1/2, 100, 3/2, 8, 1/27, 2.
Turn all 16 slips face down. Take turns flipping two. If the power and the value match, keep the pair and explain how you know ("root first, then power"). If not, flip them back.
Make your own new set of 8 pairs for the other player.

What to look for: 4^(3/2) = 8; 27^(2/3) = 9; 16^(−1/4) = 1/2; 1000^(2/3) = 100; (9/4)^(1/2) = 3/2; 32^(3/5) = 8; 81^(−3/4) = 1/27; 0.25^(−1/2) = 2. Notice that two different powers both equal 8: a value can match more than one card.

Your turn

Evaluate 4^(3/2).

Evaluate 8^(4/3).

Evaluate 9^(−3/2).

Evaluate (16/81)^(3/4).

Evaluate 1000^(2/3).

Evaluate 0.25^(−3/2).

Write 6^(2/5) in radical form in two ways.

Use a calculator: 12^(3/4), to three decimal places.

More practice: warm-up

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

Evaluate 36^(3/2).

Evaluate 32^(2/5).

Evaluate 64^(−2/3).

Write 0.5 as a fraction and evaluate 49^0.5.

More practice: on your own

Evaluate (8/125)^(−2/3).

Evaluate 16^1.25.

Evaluate 27^(2/3) − 4^(−1/2) × 2³.

Evaluate (3/4)⁻² + 0.001^(1/3).

Which is greater, 8^(2/3) or 4^(3/4)? Explain without a calculator.

Challenge

Find x if x^(2/3) = 9 (x > 0).

Show that 4^(5/2) = 2⁵ in two different ways.

A cube has a volume of V cm³. Its surface area is A = 6V^(2/3). Find A when V = 64 cm³ and explain why the exponent 2/3 makes sense.

Check your understanding

Quiz

What is 27^(2/3)?

In x^(m/n), the denominator n tells you to:

What is 16^(−3/4)?

What is (9/25)^(3/2)?

Which is equal to 100^0.5 + 8^(1/3)?

Which radical form equals 5^(3/4)?

For parents and teachers

Big idea: Expectation B1.2 (value of x^(m/n) for x > 0) and the numeric part of B1.3 (evaluating expressions with integer and rational exponents and rational bases).

How to help: Ask your teen to say the two jobs out loud: "the bottom number is the root, the top number is the power." Encourage root first. Check answers on a calculator with brackets around the exponent.

Common difficulties:

Mixing up which part is the root. 8^(2/3) is the cube root of 8 squared (4), not the square root of 8 cubed (about 22.6).
Forgetting to flip for a negative exponent, or flipping and also making the answer negative.
Decimal exponents: change 0.75 to 3/4 before evaluating.
With fractions, apply the exponent to both the numerator and the denominator.