RATIONAL EXPONENTS AND RADICALS

This is the general form of Rational Exponents and Radicals. If the power is in the form of a fraction, it is (the power) called rational exponents, and radicals if there's a square root, cube root, etcetera on the base. See the Illustration below.

Illustration:

where 'a' and 'b' are integers and 'x' is the base

As we can see in the illustration, the exponent (a/b) with the base x turned into the 'b' root of x. Note that x is just a single variable and sometimes can be used as y. The 'b' indicates the root and 'a' as power. If a variable, say "x" has power that is in the form of a fraction, it is said to be that

From the illustration,

, this read as the square root of x

Given the Illustration below is the whole set of the radical expression.

click the original photo on this site

Now, take note that "a" is a numerical coefficient, "b" is an index (like 2,3,4,5, etc.), and "c" is a radicand-that is anything under the radical symbol.

Let's have a short identification practice on Radicals.

Identify the Index, coefficient, and Radicand from the given expression below.

Coefficient: 2

Index: 3

Radicand: x

Coefficient: 1, this is when you do not see any number.

Index: 2, this is when you do not see any number as an index. It is understood that the index is 2.

Radicand: x-1

Coefficient: 1

Index: 5

Radicand: 32x^10

Coefficient: a^2

Index: 2, this is when you do not see any number as an index. It is understood that the index is 2.

Radicand: 2x+y

Since you have the idea of naming the parts of radical expression, we do believe that there's progress happening in your brain. All you have to do is keep on maintaining it. We will be discussing the operations of radicals soon. Make sure you understand the basics.

Below are the additional exercises to amplify your understanding of the subject matter.

Same thing. Identify the Index, coefficient, and Radicand from the given radical expression listed below. Please write your answers in the "post a comment" box below this page.

Coefficient: _____

Index:

Radicand:

Coefficient: _____

Index:

Radicand:

Coefficient: _____

Index:

Radicand:

Coefficient: _____

Index:

Radicand:

Coefficient: _____

Index:

Radicand:

Now, let's convert or change radicals into rational exponents. In this discussion you will expect to see all bases with an integer or fraction as powers. The square root symbol will be gone. We prepared a few illustrations for you as we guide you to go deepen the understanding with our subject.

Let's have first example.

The example above show that our coefficient is 1, the index is 3, and the radicand is x. Please go back to the uppermost Illustration on this page and check the behavior how it goes. So,

Notice that the radical symbol has gone and it was changed into rational exponent. Let's jump on the next example.

Example number 2 binomial has radicand. Take note that we have "a" to the 2nd power for our coefficient, our index is 2 (it is understood that if there's no number in radical symbol, it means that the index is 2). The radicand is a-b. Therefore, we can rewrite the expression in this manner (see below).

..and this is how it looks like. We just change the radical symbol into exponential form and the rest are just the same.

Let's have another example

From the given expression on number 3, our numerical coefficient shows 5, the index is 2, and the radicand is x to the 3rd power. Now let's convert this into exponential form, see below.

Notice that we just simply change radical into exponential for final result it would be,

Let's have another scenario. This is almost the same as number 3 but we will modify the radicand a bit.

Given that,

, here, the numerical coefficient is 5, the index is 2 and the radicand is (x+1).

Therefore,

Let us give you one more example to extend your understanding about the subject. Please follow us to guide you better.

$5.)\ -3\sqrt[3]{5}$

So, we have numerical coefficient that is -3. The index is 3 with radicand 5. In exponential form, this is

$\ -3(5)^{\frac{1}{3}}$ .

We are glad you have the basic idea and that progress will continue to the success of our objective. So, we will having a short example. We prepared a sample with partial solution. All you have to do is to supply the missing progress to satisfy the final answer. Here we go.

Find the coefficient, index, and radicand of the given expression below.

$1.) -2\ \sqrt[]{8}$

Coefficient: -2

Index: ?

Radicand: ?

$2.) \ 4\ \sqrt[3]{x+1}$

Coefficient: ?

Index: 3

Radicand: ?

$3.) \ a\ \sqrt[]{a-b}$

Coefficient: ?

Index: ?

Radicand: a-b

$4.) \ \frac{2}{3}\ \sqrt[5]{x^{10}}$

Coefficient: ?

Index: 5

Radicand: ?

$5.) \ \frac{a}{a+b}\ \sqrt[3]{(a-b)^{6}}$

Coefficient: ?

Index: 3

Radicand: ?

$6.) \ \frac{1}{3}\ m\ \sqrt[]{m+n}$

Coefficient: ?

Index: 2

Radicand: ?

$7.) \ -\frac{2}{5}\ (m-n)\ \sqrt[3]{-2(m+n)}$

Coefficient: ?

Index: 3

Radicand: ?

$8.) \ -\frac{2}{5}\ m-n\ \sqrt[3]{-2(m+n)}$

Coefficient: -n

Index: ?

Radicand: ?

$9.) \ -\frac{2m}{5n}\ + \sqrt[3]{-2(m+n)}$

Coefficient: ?

Index: ?

Radicand: -2(m+n)

$10.) \ \frac{2m}{5n}\ - \sqrt[3]{2(m-n)}$

Coefficient: -

Index: ?

Radicand: ?

DMG