Chapter Contents

• Exponents and Radicals
• 1. Simplifying Expressions with Integral Exponents
• 2. Fractional Exponents
• 3. Simplest Radical Form
• 4. Addition and Subtraction of Radicals
• 5. Multiplication and Division of Radicals (Rationalizing the Denominator)
• 6. Equations with Radicals
• Exponents and Radicals Problem Solver

• Comments, Questions?

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3. Simplest Radical Form

Before we can simplify radicals, we need to know some rules about them. These rules just follow on from what we learned in the first 2 sections in this chapter, Integral Exponents and Fractional Exponents.

Expressing in simplest radical form just means simplifying a radical so that there are no more square roots, cube roots, 4th roots, etc left to find. It also means removing any radicals in the denominator of a fraction.

Laws of Radicals

n-th root of a Number to the Power n

We met this idea in the last section, Fractional Exponents. Basically, finding the n-th root of a number is the opposite of raising the number to the power n, so they effectively cancel each other out. These 4 expressions have the same value:

The 2nd item in the equality above means:

"take the nth root first, then raise the result to the power n"

The 3rd item means:

"raise a to the power n then find the n-th root of the result"

Both steps lead back to the a that we started with.

For the simple case where n = 2, thefollowing 4 expressions all have the same value:

√a² = (√a)² = √(a²) = a

For example, if a = 9:

√9² = (√9)² = √(9²) = 9

The second item means: "Find the square root of 9 (answer: 3) then square it (answer 9)".

The 3rd item means: "Square 9 first (we get 81) then find the square root of the result (answer 9)".

We could write all this using fractional exponents as follows:

Yet another way of thinking about it is as follows:

The Product of the n-th root of a and the n-th root of b is the n-th root of ab

Example:

We could write "the product of the n-th root of a and the n-th root of b is the n-th root of ab" using fractional exponents as well:

The m-th Root of the n-th Root of the Number a is the mn-th Root of a

Example:

In words, we would say: "The 4th root of the 3rd root of 5 is equal to the 12th root of 5".

The equivalent general expression using fractional exponents is as follows:

The n-th Root of a Over the n-th Root of b is the n-th Root of a/b

(b ≠ 0)

Example:

If we write the our general expression using fractional exponents, we have:

(b ≠ 0)

Mixed Examples

Simplify the following:

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Answer

Simplest Radical Form Examples

In these examples, we are expressing the answers in simplest radical form, using the laws given above.

(a)

Answer

(b)

Answer

(c)

Answer

(d)

Answer

Exercises. Simplify:

Q1

Answer

Q2

Answer

Q3

This one requires a special trick. To remove the radical in the denominator, we need to multiply top and bottom of the fraction by the denominator.

Answer