# 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

Let's take the positive case first.

*n*-th root of a Positive Number to the Power *n*

We met this idea in the last section, Fractional Exponents.
Basically, finding the *n*-th root of a (positive) 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 powern"

The 3rd item means:

"raise

ato the powernthen find then-th root of the result"

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

For the simple case where `n = 2`, the following 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`)".

In general 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`

*n*-th root of a Negative Number to the Power *n*

We now consider the above square root example if the number `a` is negative.

For example, if `a = -5`, then:

`sqrt((-5)^2)=sqrt(25)``=5`

A negative number squared is positive, and the square root of a positive number is positive.

**In general**, we write for `a`, a negative number:

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

Notice I haven't included this part: `(sqrt(a))^2`. In this case, we would have the square root of a negative number, and that behaves quite differently, as you'll learn in the Complex Numbers chapter later.

### 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(m\ n)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)`

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

(c) `root(3)sqrt5`

(d) `sqrt7/sqrt3`

### Simplest Radical Form Examples

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

(a) `sqrt72`

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

(c) `root(3)40`

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

### Exercises. Simplify:

Q1 `sqrt(12ab^2)`

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

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.

Didn't find what you are looking for on this page? Try **search**:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Go to: Online algebra solver

### Ready for a break?

Play a math game.

(Well, not really a math game, but each game was made using math...)

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

### Share IntMath!

### Algebra Lessons on DVD

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos