Grade by Grade
B. Exponential Functions · Exponential Functions

Simplifying Expressions with Rational Exponents

B1.3About 65 minutes
Learning goals
  • I can apply the exponent laws to expressions with rational exponents.
  • I can rewrite radicals as powers, simplify, and change back to radical form.
  • I can simplify an expression first and then evaluate it.
  • I can check a simplification by substituting a value.

Words to know

exponent form
An expression written with powers, such as x^(2/3).
radical form
An expression written with root signs, such as ∛(x²).
lowest common denominator (LCD)
The smallest common denominator for adding or subtracting fractions, such as 6 for 1/2 and 1/3.
equivalent expressions
Expressions that give the same value for every allowed value of the variable.

Same laws, fraction exponents

The exponent laws do not care whether exponents are integers or fractions. That is exactly why x^(1/2) was defined as √x: so the laws would keep working. So everything from last lesson still holds, as long as the bases are positive. The only new skill is fraction arithmetic with the exponents: adding 1/2 + 1/3, subtracting 3/4 − 1/4, multiplying (2/3)(6).

1/21/3sum3/62/65/6x^(1/2) × x^(1/3) = x^(3/6 + 2/6) = x^(5/6)
Multiplying powers adds exponents, so x^(1/2) × x^(1/3) needs 1/2 + 1/3. With sixths: 3/6 + 2/6 = 5/6.

Assumption for this lesson

All variables represent positive real numbers. That way every rational power such as x^(1/2) is defined, and we never need absolute value signs.

Worked example

Example 1: Product and quotient laws

Simplify (a) x^(1/2) × x^(1/3) (b) a^(3/4) ÷ a^(1/4) (c) 6m^(2/3) × 2m^(−1/6)

  1. (a) Add exponents: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, so x^(5/6).
  2. (b) Subtract exponents: 3/4 − 1/4 = 2/4 = 1/2, so a^(1/2) = √a.
  3. (c) Coefficients: 6 × 2 = 12. Exponents: 2/3 + (−1/6) = 4/6 − 1/6 = 3/6 = 1/2. Result: 12m^(1/2) = 12√m.

x^(5/6), √a and 12√m.

Worked example

Example 2: Powers of powers with fractions

Simplify (a) (x^(2/3))⁶ (b) (16x⁸)^(3/4) (c) (27a⁶b⁻³)^(−1/3)

  1. (a) Multiply: (2/3)(6) = 4, so x⁴.
  2. (b) Each factor gets 3/4. 16^(3/4) = (∜16)³ = 2³ = 8. (x⁸)^(3/4) = x⁶. Result: 8x⁶.
  3. (c) 27^(−1/3) = 1/3. (a⁶)^(−1/3) = a⁻². (b⁻³)^(−1/3) = b¹. Result: (1/3)a⁻²b = b/(3a²).

x⁴, 8x⁶ and b/(3a²).

radical form√x × ∛(x²)exponent formx^(1/2) × x^(2/3)use the lawsx^(3/6 + 4/6) = x^(7/6)radical form⁶√(x⁷)Change roots to exponents, simplify, then change back if asked.
A reliable routine for radicals: rewrite every root as a rational exponent, simplify with the laws, then rewrite as a radical if the question asks for one.
Worked example

Example 3: Expressions with radicals

Simplify, writing the answer as a single power of x: (a) √x × ∛(x²) (b) ∜(x³) ÷ √x (c) √(x × ∛x)

  1. (a) x^(1/2) × x^(2/3) = x^(3/6 + 4/6) = x^(7/6). In radical form: ⁶√(x⁷).
  2. (b) x^(3/4) ÷ x^(1/2) = x^(3/4 − 2/4) = x^(1/4) = ∜x.
  3. (c) Inside: x × x^(1/3) = x^(4/3). Then the square root: (x^(4/3))^(1/2) = x^(2/3) = ∛(x²).

x^(7/6), x^(1/4) and x^(2/3).

Worked example

Example 4: Simplify, then evaluate

Simplify (x^(3/2)y^(−1/2))² ÷ (xy⁻³), then evaluate when x = 4 and y = 1/2.

  1. Square the numerator: x³y⁻¹.
  2. Divide by xy⁻³: x^(3 − 1) y^(−1 − (−3)) = x²y².
  3. Substitute: x²y² = 4² × (1/2)² = 16 × 1/4 = 4.
  4. Check with the original: x^(3/2) = 8 and y^(−1/2) = √2, so the numerator is (8√2)² = 128. The denominator is 4 × 8 = 32. 128 ÷ 32 = 4. ✓

x²y², which equals 4.

Worked example

Example 5: Numeric expressions

Evaluate without a calculator: (a) 2^(1/2) × 2^(3/2) (b) 5^(2/3) × 25^(2/3) (c) 9^(5/4) ÷ 9^(3/4)

  1. (a) Add exponents: 2^(4/2) = 2² = 4. Neither factor is a whole number, but the product is!
  2. (b) Same exponent, so use the power of a product law backwards: (5 × 25)^(2/3) = 125^(2/3) = 5² = 25.
  3. (c) Subtract exponents: 9^(2/4) = 9^(1/2) = 3.

4, 25 and 3.

Just for fun

Exponent laws are a lot like getting dressed for a Canadian winter. Multiplying? Add layers. Dividing? Take some off. Power of a power? Three jackets, each with two sweaters underneath: that's 3 × 2 = 6 layers, so you multiply. Either way, you can't move your arms.

Hands-on activity

Prove it with a calculator: substitution checks

You need: a calculator, paper and a pencil.

  1. Simplify √x × ∛x to a single power of x.
  2. Choose x = 5. Calculate the original expression √5 × ∛5 on the calculator (5 ^ (1 ÷ 2) × 5 ^ (1 ÷ 3)), and then your simplified answer with x = 5. Do they match?
  3. Repeat with x = 2 and x = 10.
  4. Now test a wrong simplification on purpose: is √x × ∛x equal to x^(1/6)? Try x = 5.
  5. Make up your own expression with two roots, simplify it, and check it the same way.

What to look for: √x × ∛x = x^(5/6). For x = 5, both sides are about 3.824. For x = 2, about 1.782; for x = 10, about 6.813. The wrong answer x^(1/6) gives about 1.308 for x = 5, so it fails. One mismatch proves an expression wrong; matches at several values make you confident it is right (the laws prove it).

Your turn

1

Simplify x^(2/5) × x^(1/5).

2

Simplify y^(5/6) ÷ y^(1/3).

3

Simplify (a^(3/4))⁸.

4

Simplify (64x⁹)^(1/3).

5

Write √x × ∜x as a single power of x.

6

Simplify (9m⁴n⁻²)^(−1/2).

7

Evaluate 3^(1/3) × 9^(1/3).

More practice: warm-up

1

Simplify x^(1/2) × x^(1/2).

2

Simplify (x^(1/3))³.

3

Simplify a^(7/4) ÷ a^(3/4).

4

Write ∛(x⁴) as a power.

5

Write x^(3/2) in radical form.

More practice: on your own

1

Simplify (4x^(2/3))(−3x^(1/2)).

2

Simplify (25a⁴b⁻⁶)^(3/2).

3

Simplify ∛x ÷ x² as a single power, then write it with a positive exponent.

4

Simplify (x^(1/2)y^(1/3))⁶ ÷ (xy).

5

Evaluate 16^(3/8) × 16^(1/8).

Challenge

1

Simplify √(x√(x√x)).

2

Simplify (8^(2/3) × a^(1/2))² ÷ (4a)^(1/2). Write the answer with exponents.

3

Find k so that x^(1/4) × x^k = ∛x.

Check your understanding

Quiz

1.

Simplify x^(1/3) × x^(1/2).

2.

Simplify (x^(3/4))^(4/3).

3.

Simplify (8x⁶)^(2/3).

4.

Which is equal to √x ÷ ∛x?

5.

Evaluate 2^(3/2) × 2^(1/2).

6.

Simplify (16a⁴)^(−1/2).

For parents and teachers

Big idea: Expectation B1.3: simplify algebraic expressions containing rational exponents, and evaluate numeric expressions with rational exponents.

How to help: Fraction skills carry this lesson. If your teen struggles, practise adding and subtracting fractions with unlike denominators for five minutes first. Encourage the substitution check from the activity: plugging in a number like x = 5 catches most mistakes.

Common difficulties:

  • Adding fraction exponents by adding tops and bottoms: 1/2 + 1/3 is 5/6, not 2/5.
  • Applying a rational exponent to the coefficient incorrectly: (16a⁴)^(1/2) = 4a², not 8a² or 16a².
  • Leaving a mix of radicals and exponents. Convert everything to exponent form first.