Fractional Exponents and Radicals

Discover why raising a number to a fractional power is identical to taking a root, derive the general rule for any fractional exponent, and learn the tip for evaluating expressions by taking the root before the power.

Lesson 2.4 of Number Systems & Operations in Math 10C: Alberta curriculum lessons.

What you'll learn in this lesson

• The Connection Between Fractional Exponents and Radicals. Discover why raising a number to a fractional power is identical to taking a root, derive the general rule for any fractional exponent, and learn the tip for evaluating expressions by taking the root before the power.
• Converting Between Forms. Practice translating expressions from fractional exponent form to radical form and back, using the rule that the denominator becomes the index and the numerator becomes the power.
• Evaluating Expressions with Fractional Exponents. Evaluate expressions step-by-step without a calculator, handle negative bases and even powers, and apply fractional exponents to a real-world brain-mass formula.

Inside the lesson: a free preview

Discovering the Pattern

What Does x^(1/2) Mean?. Let's see what happens when we raise numbers to the exponent (1)/(2). Look carefully at the results below and compare them to the square root of each number.

Observation. Raising a number to the exponent (1)/(2) gives the same result as taking its square root.

What Does x^(1/3) Mean?. Now let's try the exponent (1)/(3). Compare the results to the cube root of each number.

Observation. Raising a number to the exponent (1)/(3) gives the same result as taking its cube root.

The Rule: x^(1/n) = √(x)

Specific Cases. x^(1/2) = √(x), square root

x^(1/3) = √(x), cube root

x^(1/4) = √(x), fourth root

x^(1/5) = √(x), fifth root

Why Does This Work?. Think about what the exponent rules require. We know that (x^a)^b = x^(a · b).

If x^(1/2) is squared, we get (x^(1/2))^2 = x^(1/2 · 2) = x^1 = x.

What other operation, when applied twice, gives you back the original number?

The square root! So x^(1/2) must equal √(x).

What About Exponents Like (2)/(3) or (3)/(4)?

The General Rule. When the numerator of the fractional exponent is something other than 1, the numerator becomes the power and the denominator becomes the root (the index of the radical).

Two Equivalent Forms. Both forms give the same answer. You can:

Form 1: Raise to the power first, then take the root: √(x^a)

Form 2: Take the root first, then raise to the power: (√(x))^a

For example: x^(2/3) = √(x^2) = (√(x))^2

And: x^(3/4) = √(x^3) = (√(x))^3

Strategy Tip: Take the Root First. It is usually easier to take the root first, then raise to the power. Taking the root first keeps the numbers smaller.

For instance, to evaluate 8^(2/3):

Take the root first: √(8) = 2

Then raise to the power: 2^2 = 4

Much easier than computing 8^2 = 64 first and then finding √(64).

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