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:

`root(n)(a^n)=(root(n)a)^n``=root(n)((a^n))=a`

The 2nd item in the equality above means:

"take the n-th 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:

`sqrt(a^2)=(sqrt(a))^2``=sqrt((a^2))=a`

For example, if a = 9:

`sqrt(9^2)=(sqrt(9))^2``=sqrt((9^2))=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:

`root(n)(a^n)=(a^(1//n))^n``=(a^n)^(1//n)=a`

Yet another way of thinking about it is as follows:

`(a^(1/n))^n=a^((1/nxxn))=a`

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

`root(n)axxroot(n)b=root(n)(ab)`

Example:

`root(4)7xxroot(4)5=root(4)(7xx5)=root(4)35`

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:

`a^(1//n)xxb^(1//n)=(ab)^(1//n)`

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

`root("m")(root("n")a)=root("mn")a`

We could write this as:

`(a^(1//n))^(1//m)=(a)^(1//(mn))`

Example:

`root(4)(root(3)5)=root(12)5`

This has the same meaning:

`(5^(1//3))^(1//4)=(5)^(1//(12))`

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

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

`root(n)a/root(n)b=root(n)(a/b)`(`b ≠ 0`)

Example:

`root(3)375/root(3)3=root(3)(375/3)``=root(3)125=5`

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

`a^(1//n)/b^(1//n)=(a/b)^(1//n)` (`b ≠ 0`)

Mixed Examples

Simplify the following:

(a) `root(5)(4^5)`

Answer

(b) `root(3)2root3(3)`

Answer

(c) `root(3)sqrt5`

Answer

(d) `sqrt7/sqrt3`

Answer

Simplest Radical Form Examples

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

(a) `sqrt72`

Answer

(b) `sqrt(a^3b^2)`

Answer

(c) `root(3)40`

Answer

(d) `root(5)(64x^8y^(12))`

Answer

Exercises. Simplify:

Q1 `sqrt(12ab^2)`

Answer

Q2 `root(4)(64r^3s^4t^5`

Answer

Q3 `sqrt(x/(2x+1)`

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