Rational Exponents
Lesson 3: Rational Exponents
Evaluate and simplify powers with rational exponents.
Rational exponents (fractional exponents) are exponents that are fractions or rational expressions.
A rational exponent can be converted to its equivalent radical form.
Exponent laws can be applied to rational exponents in the same way they work for any other types of exponents.
In general, in order to convert a rational exponent into a radical, keep the base as a radicand raised to the numerator exponent; place the denominator as an index of the radical symbol (root); place the coefficient in front of the radical symbol.
Multiplying and Dividing Expressions Containing Powers with Rational Exponents
In order to multiply expressions containing powers with rational exponents, we need to first multiply coefficients, then keep the corresponding base and add the corresponding exponents. Since the exponents are rational, a common denominator would need to found.
In order to divide expressions containing powers with rational exponents, we need to first divide or reduce coefficients, then keep the corresponding base and subtract the corresponding exponents. Since the exponents are rational, a common denominator would need to found.
Power of a Power Exponent Law with Rational Exponents
In order to raise an expression containing powers with rational exponents, we need to every part of that expression to the outside rational exponent. If there is a coefficient, it need to be raised to the outside rational exponent as per common rule raising any base to an exponent. For the powers containing rational exponents, the bases would have to remain the same, but the exponents would multiply by the outside rational exponent.
No matter what operation is performed on the expressions containing powers with rational exponents, it is important to follow exponent laws, pay attention to which part of the expression is raised to the exponent and which one is not (separate a coefficient), pay attention to signs (positive-negative).
Watch how to do operations on expressions containing powers with rational exponents:
Key ideas: Rational Exponents
am/n = (ⁿ√a)m = ⁿ√(am). The denominator n is the root, and the numerator m is the power. All the exponent laws work with fractional exponents, and a negative exponent still means “reciprocal”.
To solve an exponential equation, try to write both sides with the same base, then set the exponents equal.
Worked examples
Example 1
Evaluate a) 323/5 b) 25−1/2
- a) ⁵√32 = 2, and 23 = 8.
- b) 251/2 = 5, so 25−1/2 = 1/5.
Example 2
Simplify a) (x2/3)3/4 b) (27x6)2/3
- a) Multiply the exponents: x(2/3)(3/4) = x1/2 = √x.
- b) 272/3 = (∛27)2 = 9 and (x6)2/3 = x4: 9x4.
Example 3
Solve 8x = 32.
- Write both sides as powers of 2: (23)x = 25, so 23x = 25.
- 3x = 5, so x = 5/3.
- Check: 85/3 = (∛8)5 = 25 = 32 ✓
Common mistakes
- Swapping the root and the power: 323/5 means the 5th root, not the cube root.
- Applying the exponent to the variable but not the coefficient: (27x6)2/3 is 9x4, not 27x4.
- Treating a−1/2 as negative; it is 1/√a.
Questions students ask
Is (−8)1/3 defined?
Yes, it is −2, because odd roots of negative numbers are real. (−4)1/2 is not a real number.
Why write radicals as rational exponents?
The exponent laws then do the simplifying for you, and it prepares you for exponential functions.
Check your understanding
Practise with the Grade 11 quizzes. You’ll see right away if each answer is correct.