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Fractional Powers

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Are you aware that powers or exponents may not be whole numbers but fractions? Yes, exponents also exist as fractions and we would be discussing on them herein.

In this article, we will see what fractional powers are, what negative fractional powers are, their rules, and examples of application.

**Fractional powers** or fraction exponents are expressions which are being powered by fractions and are in the form **x**^{a/b} .

We are more familiar with whole-number exponents in the form x^{a}. Because ** x** has been powered by

.

Solve for .

**Solution**

Solve for

Solution

The power of a fraction in decimal form is an exponent which is a fraction that is expressed as a decimal. It occurs in the form;

,

where ** a** and

Remember that ** a** and

Solve for .

**Solution**

Recall that;

Then;

Recalling that , we then have

In conclusion,

Negative fractional powers occur when an expression has been powered by a negative fraction. This appears in the form **x ^{–}**

.

This is in line with **the rule of negative exponents** which states that

.

The negative fractional powers is among the rules of fractional powers which shall be discussed below.

These rules when applied would enable you easily solve fractional exponents problems. However, before going to the rules note that fractional powers are defined by the form

as well as

With the knowledge of this definition, the following rules should be applied.

**Rule 1:** When the base for instance* *** x** is powered by a negative fraction for example , find the

Solve .

**Solution**

By applying rule 1,

**Rule 2:** When the base is a fraction for instance* *** ,** and is powered by a negative fraction for example , find the

Solve

**Solution**

By applying rule 2,

**Rule 3:** When the product of two or more fractional powers in this case, and , have the same base in this case * x*, then find the

Solve .

**Solution**

By applying rule 3,

**Rule 4:** When the product of two or more fractional powers in this case, and , have the same base in this case * x*, then find the

Solve

**Sol****ution**

By applying rule 4,

**Rule 5:** When the quotient of two unit-fractional powers in this case, and , have the same base in this case * x*, then find the

Solve

**Solution**

By applying rule 5,

**Rule 6:** When the quotient of two fractional powers in this case, and , have the same base in this case * x*, then find the

Solve .

**Solution**

By applying rule 6,

**Rule 7:** When the product of two fractional powers have different bases in this case ** x** and

Solve .

**Solution**

By applying rule 7,

**Rule 8:** When the quotient of two fractional powers have different bases in this case ** x** and

Solve .

**Solution**

By applying rule 8,

Solve the following;

a.

b.

c.

**Solution**

a.

The first thing to do is to see if you can change the number to exponent form (indices).

Note that;

Therefore;

Recall that;

Then;

b.

Recall that;

Then;

c.

The first thing to do is to see if you can change the number to exponent form (indices).

Therefore;

Recall that;

Then;

or you could solve directly from this point;

How is a binomial expansion for fractional powers done?

The binomial expansion for fractional powers is carried out simply by applying the formula

where **n** is the power or exponent.

Solve for the first 4 terms of .

**Solution**

Ensure you factorise or re-express the expression bearing the exponent to conform to the form;

.

So, your plan is to convert (8 + 2y) to (1 + y). To achieve that, factorise 8 + 2y by 8. You would have

Let

Substitute into the equation

Recalling that , we then have

Recall that

Also, we are only interested in the first 4 terms, therefore;

Substitute the real value of ** a **as;

Therefore;

And so

Some more examples would give you a better understanding of fractional powers.

If the cube root of a number is squared and the result is 4. Find the number.

**Solution**

Let the unknown number be y. So the cube root of a number, y being square and resulting to 4 is expressed as .

Note that

ThenTake the reciprocal of the roots in both sides. The reciprocal of is, therefore;

Recall that

So,

- Fractional powers or fraction exponents are expressions which are being powered by fractions and are in the form
**x**^{a/b}. - Negative fractional powers occurs when an expression has been powered by a negative fraction.
- Fractional power rules when applied would enable you easily solve fractional exponents problems.
- The binomial expansion for fractional powers is carried out simply by applying the formula;

You integrate expressions with fractional powers by simply applying the rules of integral calculus.

You calculate fractional powers by applying the rules of fractional powers.

More about Fractional Powers

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