# data[].attributes.attachment_link.title
Evaluating Numbers with Rational Exponents by using Radical Notation - Basic Example 1

# data[].attributes.attachment_link.title
Evaluating Expressions with Negative Exponents from Thinkwell College Algebra

# data[].attributes.attachment_link.title
Graphing Radical Functions & Equations, Transformations, Domain & Rangle, Plotting Points, Algebra

# data[].attributes.attachment_link.title
Ex: Find the Inverse of a Square Root Function with Domain and Range

# data[].attributes.attachment_link.title
How to find domain and range of a radical function 1.2 Precalculus mathgotserved

# data[].attributes.attachment_link.title
Simplifying Expressions with Rational Exponents

# data[].attributes.attachment_link.readable_text
Also called "Radicals" or "Rational Exponents"

Whole Number Exponents

First, let us look at whole number exponents:

The exponent of a number says how many times to use the number in a multiplication.

In this example: 8² = 8 × 8 = 64

Fractional Exponents

But what if the exponent is a fraction?

An exponent of 1 2 is a square root

An exponent of 1 3 is a cube root

An exponent of 1 4 is a 4th root

And so on!

Why?

Let's see why in an example.

First, the Laws of Exponents tell us how to handle exponents when we multiply:

Which shows that x²x³ = x⁽²⁺³⁾ = x⁵

So let us try that with fractional exponents:

So times itself gives 9.

Now, what do we call a number that, when multiplied by itself, gives another number? The square root of that other number!

See:

√9 × √9 = 9

And:

Try Another Fraction

Let us try that again, but with an exponent of one-quarter (1/4):

Example:

So used 4 times in a multiplication gives 16,

and so 16¼ is a 4th root of 16

General Rule

It worked for ½, it worked with ¼, in fact it works generally:

In other words:

A fractional exponent like 1/n means to take the n-th root:

Example: What is 271/3 ?

What About More Complicated Fractions?

What about a fractional exponent like ?

That is really saying to do a cube (3) and a square root (1/2), in any order.

Let me explain.

A fraction (like m/n) can be broken into two parts:

So, because m/n = m × (1/n) we can do this:

The order does not matter, so it also works for m/n = (1/n) × m:

And we get this:

A fractional exponent like m/n means:

Some examples:

Example: What is 43/2 ?

or

Either way gets the same result.

Example: What is 274/3 ?

or

It was certainly easier the 2nd way!

Now ... Play With The Graph!

See how smoothly the curve changes when you play with the fractions in this animation, this shows you that this idea of fractional exponents fits together nicely:

Things to try:

Start with m=1 and n=1, then slowly increase n so that you can see 1/2, 1/3 and 1/4

Then try m=2 and slide n up and down to see fractions like 2/3 etc

Now try to make the exponent −1

Lastly try increasing m, then reducing n, then reducing m, then increasing n: the curve should go around and around

321,322, 1089, 1090, 1091, 1092, 2266, 3997, 17, 2267

# data[].attributes.attachment_link.description
The exponent of a number says how many times to use the number in a multiplication. So what does a fractional exponent mean?

# data[].attributes.attachment_link.readable_text
Radical Equations

Solving Radical Equations

We can get rid of a square root by squaring. (Or cube roots by cubing, etc)

But Warning: this can sometimes create "solutions" which don't actually work when we put them into the original equation. So we need to Check!

Follow these steps:

isolate the square root on one side of the equation

square both sides of the equation

Then continue with our solution!

Example: solve √(2x+9) − 5 = 0

Now it should be easier to solve!

Check: √(2·8+9) − 5 = √(25) − 5 = 5 − 5 = 0

That one worked perfectly.

More Than One Square Root

What if there are two or more square roots? Easy! Just repeat the process for each one.

It will take longer (lots more steps) ... but nothing too hard.

Example: solve √(2x−5) − √(x−1) = 1

We have removed one square root.

Now do the "square root" thing again:

We have now successfully removed both square roots.

Let us continue on with the solution.

It is a Quadratic Equation! So let us put it in standard form.

Using the Quadratic Formula (a=1, b=−14, c=29) gives the solutions:

2.53 and 11.47 (to 2 decimal places)

Let us check the solutions:

2.53: √(2×2.53−5) − √(2.53−1) ≈ −1 Oops! Should be plus 1.

11.47: √(2×11.47−5) − √(11.47−1) ≈ 1 Yes that one works.

There is really only one solution:

Answer: 11.47 (to 2 decimal places)

See? This method can sometimes produce solutions that don't really work!

The root that seemed to work, but wasn't right when we checked it, is called an "Extraneous Root"

So: Checking is important.

# data[].attributes.attachment_link.description
We can get rid of a square root by squaring. (Or cube roots by cubing, etc) But Warning: this can sometimes create "solutions" which don't actually work when we put them into the original equation. So we need to Check!

# data[].attributes.attachment_link.readable_text
Exponents are also called Powers or Indices

The exponent of a number says how many times to use the number in a multiplication.

In words: 8 ² could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

Try it yourself:

So an Exponent saves us writing out lots of multiplies!

a⁷ = a × a × a × a × a × a × a = aaaaaaa

Notice how we wrote the letters together to mean multiply? We will do that a lot here.

The Key to the Laws

Writing all the letters down is the key to understanding the Laws

Which shows that x²x³ = x⁵, but more on that later!

So, when in doubt, just remember to write down all the letters (as many as the exponent tells you to) and see if you can make sense of it.

All you need to know ...

The "Laws of Exponents" (also called "Rules of Exponents") come from three ideas:

The exponent says how many times to use the number in a multiplication.

A negative exponent means divide, because the opposite of multiplying is dividing

If you understand those, then you understand exponents!

And all the laws below are based on those ideas.

Laws of Exponents

Here are the Laws (explanations follow):

Law

Example

x ¹ = x

6 ¹ = 6

x ⁰ = 1

7 ⁰ = 1

x⁻¹ = 1/x

4⁻¹ = 1/4

x mx n = x m+n

x ²x ³ = x ²⁺³ = x ⁵

x m/x n = x m-n

x ⁶/x ² = x ⁶⁻² = x ⁴

(x m) n = x mn

(x ²) ³ = x 2×3 = x ⁶

(xy) n = x ny n

(xy) ³ = x ³y ³

(x/y) n = x n/y n

(x/y) ² = x ² / y ²

x-n = 1/x n

x⁻³ = 1/x ³

And the law about Fractional Exponents:

Laws Explained

The first three laws above (, and ) are just part of the natural sequence of exponents. Have a look at this:

Example: Powers of 5

.. etc..

.. etc..

Look at that table for a while ... notice that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or 5 times smaller) depending on whether the exponent gets larger (or smaller).

With x mx n, how many times do we end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.

Like the previous example, how many times do we end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because we are dividing), for a total of "m-n" times.

(Remember that x/ x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)

This law can also show you why x⁰=1 :

First you multiply "m" times. Then you have to do that "n" times, for a total of m×n times.

Example: (x³)⁴ = (xxx)⁴ = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x¹²

To show how this one works, just think of re-arranging all the "x"s and "y"s as in this example:

Similar to the previous example, just re-arrange the "x"s and "y"s

The law that

OK, this one is a little more complicated!

I suggest you read Fractional Exponents first, or this may not make sense.

Anyway, the important idea is that:

And so a fractional exponent like is really saying to do a cube (3) and a square root (1/2), in any order.

Just remember from fractions that m/n = m × (1/n):

Example:

The order does not matter, so it also works for m/n = (1/n) × m:

Example:

Exponents of Exponents ...

What about this example?

We do the exponent at the top first, so we calculate it this way:

And That Is It!
If you find it hard to remember all these rules, then remember this:
you can work them out when you understand the
three ideas near the top of this page

Oh, One More Thing ... What if x = 0?

The Strange Case of 0⁰

There are different arguments for the correct value of 0 ⁰

0 ⁰ could be 1, or possibly 0, so some people say it is really "indeterminate":

x ⁰ = 1, so ...

0 ⁰ = 1

0 n = 0, so ...

0 ⁰ = 0

When in doubt ...

0 ⁰ = "indeterminate"

# data[].attributes.attachment_link.description
Exponents are also called Powers or Indices The exponent of a number says how many times to use the number in a multiplication. In words: 8 2 could be called "8 to the second power", "8 to the power 2" or simply "8 squared" Try it yourself: So an Exponent saves us writing out lots of multiplies!

# data[].attributes.attachment_link.readable_text
To simplify a square root: make the number inside the square root as small as possible (but still a whole number):

Example: √12 is simpler as 2√3

Get your calculator and check if you want: they are both the same value!

Here is the rule: when a and b are not negative

And here is how to use it:

Example: simplify √12

12 is 4 times 3:

Use the rule:

And the square root of 4 is 2:

So √12 is simpler as 2√3

Another example:

Example: simplify √8

√8 = √(4×2) = √4 × √2 = 2√2

(Because the square root of 4 is 2)

And another:

Example: simplify √18

√18 = √(9 × 2) = √9 × √2 = 3√2

It often helps to factor the numbers (into prime numbers is best):

Example: simplify √6 × √15

First we can combine the two numbers:

Then we factor them:

√(6 × 15) = √(2 × 3 × 3 × 5)

Then we see two 3s, and decide to "pull them out":

√(2 × 3 × 3 × 5) = √(3 × 3) × √(2 × 5) = 3√10

Fractions

There is a similar rule for fractions:

Example: simplify √30 / √10

First we can combine the two numbers:

Then simplify:

Some Harder Examples

See if you can follow the steps:

Example: simplify 2√12 + 9√3

First simplify 2√12:

Now both terms have √3, we can add them:

4√3 + 9√3 = (4+9)√3 = 13√3

Surds

Note: a root we can't simplify further is called a Surd. So √3 is a surd. But √4 = 2 is not a surd.

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.