B. Exponential Functions · Exponential Functions

Evaluating Expressions with Integer and Rational Exponents

B1.2 B1.1 B1.5About 70 minutes

Learning goals

I can use the exponent rules with fraction exponents, such as 2^(1/2) × 2^(3/2) = 2².
I can evaluate expressions that mix integer and rational exponents and rational bases, using the order of operations.
I can evaluate powers like 1.01¹²⁰ with a calculator and explain what they mean in a real situation.
I can compare powers by estimating or by rewriting them with the same base or the same exponent.

Words to know

order of operations
Brackets, then exponents, then multiplication and division (left to right), then addition and subtraction (left to right). Often remembered as BEDMAS.

common base
Writing different bases as powers of one number, such as 4 = 2² and 8 = 2³, so the exponent rules can be used.

growth factor
The number a quantity is multiplied by in each time step. A 1% increase has growth factor 1.01.

exact value
An answer written without rounding, such as 27/8 or √2.

One percent, many times

Suppose something grows by 1% every month, like money in a savings plan or the number of members in an online club. After one month it has been multiplied by 1.01. After 10 years (120 months) it has been multiplied by 1.01 a total of 120 times: 1.01¹²⁰.

No one wants to multiply 120 times by hand. With a calculator, 1.01¹²⁰ ≈ 3.30. So 1% a month more than triples the amount in 10 years. Evaluating expressions like this, correctly and with confidence, is the skill for this lesson.

The exponent rules still work with fractions

The product, quotient and power of a power rules work for all rational exponents (with positive bases):

2^(1/2) × 2^(3/2) = 2^(1/2 + 3/2) = 2² = 4 (even though √2 and 2^(3/2) are not whole numbers!)
32^(3/5) ÷ 32^(1/5) = 32^(2/5) = (⁵√32)² = 4
(5^(1/3))⁶ = 5^(6/3) = 5² = 25

Often the smart move is to simplify first and evaluate last.

Worked example

Example 1: Simplify, then evaluate

Evaluate (a) 6^(3/4) × 6^(5/4) (b) 7^(5/3) ÷ 7^(2/3) (c) (10^(2/3))^(3/2) (d) (4^(1/3))² × 4^(4/3)

(a) Add exponents: 3/4 + 5/4 = 8/4 = 2. 6² = 36.
(b) Subtract: 5/3 − 2/3 = 3/3 = 1. 7¹ = 7.
(c) Multiply: (2/3)(3/2) = 1. 10¹ = 10.
(d) (4^(1/3))² = 4^(2/3). Then 4^(2/3) × 4^(4/3) = 4^(6/3) = 4² = 16.

36, 7, 10, 16

Mixed expressions: order of operations

When powers are added or subtracted, there is no exponent rule to combine them. Evaluate each power first (exponents come before addition in BEDMAS), then add or subtract. Fraction answers are usually best left as exact fractions.

Evaluate each power separately, then combine. You cannot add the exponents of powers that are being added.

Worked example

Example 2: A mixed expression with rational bases

Evaluate (2/3)⁻² + (8/27)^(1/3) − 4^(−3/2).

(2/3)⁻² = (3/2)² = 9/4.
(8/27)^(1/3) = ∛8 / ∛27 = 2/3.
4^(−3/2) = 1 / (√4)³ = 1/8.
Common denominator 24: 9/4 + 2/3 − 1/8 = 54/24 + 16/24 − 3/24 = 67/24.
Calculator check: 2.25 + 0.6667 − 0.125 = 2.7917, and 67 ÷ 24 = 2.7917. ✓

67/24 (≈ 2.79)

Worked example

Example 3: Use a common base

Evaluate 4^(3/2) × 8^(−1/3) by writing both as powers of 2.

4 = 2², so 4^(3/2) = (2²)^(3/2) = 2³.
8 = 2³, so 8^(−1/3) = (2³)^(−1/3) = 2⁻¹.
2³ × 2⁻¹ = 2² = 4.
Check directly: 4^(3/2) = 8 and 8^(−1/3) = 1/2, and 8 × 1/2 = 4. ✓

Worked example

Example 4: Powers of decimals in context

Evaluate to three decimal places and explain each in words: (a) 1.01¹²⁰ (b) 0.95¹⁰ (c) 1.06^(1/12)

(a) 1.01¹²⁰ ≈ 3.300. Growing 1% per period for 120 periods multiplies the amount by about 3.3.
(b) 0.95¹⁰ ≈ 0.599. Losing 5% per period for 10 periods leaves about 59.9% of the original amount (not 50%!).
(c) 1.06^(1/12) ≈ 1.005. If something grows 6% in a year, the matching monthly growth factor is the 12th root of 1.06, about 1.004 87, which is a little less than 0.5% a month.

3.300, 0.599, 1.005

The value of $1 growing at 1% per month: y = 1.01ˣ. After 48 months it is about $1.61, and after 120 months about $3.30. You will study this kind of growth with money in the Financial Mathematics unit.

Comparing powers without a calculator

Same base, base > 1: the bigger exponent wins. 3^(3/4) < 3^(4/5) because 3/4 < 4/5.
Same base, base between 0 and 1: the bigger exponent gives the smaller number. 0.5³ < 0.5².
Different bases and exponents: raise both to a power that clears the fractions. To compare 2^(1/2) and 3^(1/3), raise both to the 6th power: (2^(1/2))⁶ = 2³ = 8 and (3^(1/3))⁶ = 3² = 9. Since 8 < 9, 2^(1/2) < 3^(1/3).

Just for fun

Funny but true: if you could fold a sheet of paper 0.1 mm thick in half 42 times, its thickness would be 0.1 × 2⁴² mm, which is about 440 000 km. That is farther than the distance from Earth to the Moon (about 384 000 km on average)! In real life, an ordinary sheet gets too thick to fold after only a handful of folds.

Hands-on activity

Calculator detective: one expression, three entries

You need: a scientific calculator (or a free calculator app) and a pencil.

Evaluate 27^(2/3) three ways: (a) 27 ^ ( 2 ÷ 3 ); (b) first find 27 ^ ( 1 ÷ 3 ), then square the answer; (c) first square 27, then take the cube root of the result with 729 ^ ( 1 ÷ 3 ).
Now type the wrong entry 27 ^ 2 ÷ 3. Write down what the calculator really did.
Evaluate 1.01 ^ 120 and 1.1 ^ 12. Describe a real situation that each one could model, including the length of time.
Compare 2^(1/2) and 3^(1/3) on the calculator to confirm the tip box.

What to look for: entries (a), (b) and (c) all give 9. The wrong entry gives 27² ÷ 3 = 243. 1.01¹²⁰ ≈ 3.30 models 1% per month for 120 months (10 years); 1.1¹² ≈ 3.14 models 10% per period for 12 periods (12 years if the periods are years). Finally 2^(1/2) ≈ 1.414 and 3^(1/3) ≈ 1.442, so 3^(1/3) is larger.

Your turn

Simplify, then evaluate: 3^(1/2) × 3^(3/2)

Simplify, then evaluate: 8^(5/3) ÷ 8^(4/3)

Simplify, then evaluate: (11^(1/4))⁸

Evaluate: 25^(1/2) − 27^(1/3)

Evaluate: 16^(−1/2) + (1/2)⁻¹

Evaluate: 9^(3/2) × 9⁻¹

Use a calculator: 1.02²⁴ (three decimal places).

Use a calculator: 0.8⁵ (three decimal places). What percent of an amount is left after five 20% decreases?

More practice: warm-up

Simplify: 5^(1/3) × 5^(2/3)

Simplify: (2^(1/2))⁴

Evaluate: 4^(1/2) + 4⁰

Evaluate: 1.5²

Which is bigger, 7^(1/2) or 7^(1/3)?

More practice: on your own

Evaluate: (1/4)^(−1/2) × (1/8)^(1/3)

Evaluate exactly: 0.25^(1/2) + 0.008^(1/3)

Evaluate by writing with base 3: 9^(3/4) × 27^(1/2)

Evaluate: (−2)³ + 16^(3/4) − 2⁻²

A video's views grow about 15% per day. Evaluate 1.15⁷ and explain what it means.

Challenge

Which is larger: 5^(1/2) or 11^(1/3)? Decide without a calculator.

Evaluate exactly: [(27/8)^(2/3) − (9/4)^(1/2)] ÷ (1/2)²

Show that 1.06^(1/12), raised to the 12th power, gives back 1.06. Then explain why a 6% yearly increase is NOT the same as 0.5% every month.

Check your understanding

Quiz

Simplify 2^(3/4) × 2^(5/4).

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

Which is closest to 1.03¹⁰?

Evaluate (9^(1/3))^(3/2).

Evaluate 8^(2/3) × 4^(−1/2).

Something loses 5% of its value each year. What fraction of the value is left after 10 years?

For parents and teachers

Big idea (B1.2, with B1.1 and B1.5): students evaluate numerical expressions that mix integer and rational exponents and rational bases, with and without technology, using the exponent rules and the order of operations. The example 1.01¹²⁰ comes straight from the kind of expression the curriculum asks students to evaluate.

How to help: Before your teen presses = on the calculator, ask "About how big should this be?" Ask them to explain what 0.95¹⁰ means in words ("what is left after ten 5% drops").

Common difficulties:

Adding exponents when powers are added: 16^(1/2) + 16^(1/4) is 6, not 16^(3/4).
Thinking ten 5% losses is a 50% loss: 0.95¹⁰ ≈ 0.60, so about 40% is lost.
Fraction arithmetic: many errors in mixed expressions are common-denominator mistakes, not exponent mistakes.
Calculator brackets around fraction exponents and negative bases.