Fractional Powers

Understanding Fractional Powers and How They Work

In mathematics, exponents or powers can be more than just whole numbers - they can also be fractions. This article will explore the concept of fractional powers, including their definition, rules, and practical uses.

So what exactly are fractional powers? They are expressions written as xa/b, where x is raised to the power of a fraction. While we are familiar with whole-number exponents (xa, where x is multiplied by itself a times), a fraction as an exponent indicates taking the root of the expression. For example, x1/a would involve finding the a root of x.

You may also come across fractional powers expressed in decimal form, such as a.b. It's important to remember that a and b represent the digits of the decimal number. For instance, in the decimal 3.2, a = 3 and b = 2.

But what about negative fractional powers? These occur when an expression is raised to the power of a negative fraction, represented as x–a/b. In such cases, the reciprocal of the expression is raised to the fraction, following the rule of negative exponents which states that x–a equals 1/xa.

Applying the Rules of Fractional Powers

To effectively solve problems involving fractional powers, there are several rules that can be applied. These rules are based on the definition of fractional powers, which states that xa/b = .

Let's take a look at these rules and how they can be used:

• Rule 1: When the base (x) is raised to a negative fraction (–a/b), take the b root of x and raise it to the power of a, then find the reciprocal of the result.
• Rule 2: When the base is a fraction (a/b) and is raised to a negative fraction (–a/b), take the b root of a and raise it to the power of a.
• Rule 3: When two or more fractional powers have the same base (x), take the ab root of x and raise it to the sum of b and a.
• Rule 4: When the product of two fractional powers have the same base (x) but different powers (bm and an), take the ab root of x and raise it to the sum of bm and an.
• Rule 5: When the quotient of two unit-fractional powers have the same base (x) but different powers (b and a), take the ab root of x and raise it to the difference of b and a.
• Rule 6: When the quotient of two fractional powers have the same base (x) but different powers (bm and an), take the ab root of x and raise it to the difference of bm and an.
• Rule 7: When the product of two fractional powers have different bases (x and y) but the same power (a), take the a root of xy.
• Rule 8: When the quotient of two fractional powers have different bases (x and y) but the same power (a), take the a root of .

Now, let's apply these rules to some examples:

• a.
• b.
• c.

Solutions:

• a. =
• b. =
• c. = or (directly solving without converting to exponent form)

Using Binomial Expansion for Fractional Powers

Another useful concept when dealing with fractional powers is binomial expansion. It involves using the formula where n is the power or exponent.

Let's try an example to see this in action:

Simplify the first 4 terms of .

Solution:

By using the formula, we get:

=

=

=

=

Using Factorization in Calculating Fractional Powers

To better understand fractional powers, it can be helpful to factorize or re-express the expression in a form that follows the pattern (1 + y). For instance, if our goal is to convert (8 + 2y) to (1 + y), we can factorize 8 + 2y by 8 to get:

(1 + y) = 8(1 + y)

We can then substitute this into the original equation:

(8 + 2y) = (1 + y)

Recalling that y = 1 + y - 1, we can rewrite the equation as:

(8 + 2y) = (8 + y - 8)

Since we only need the first four terms, we can substitute the value of a as:

a = 8 + y - 8

Thus, the final result is:

(1 + y) = (8 + y - 8)

And so, (8 + 2y) has successfully been converted to (1 + y).

More Examples in Calculating Fractional Powers

Now, let's look at some additional examples to further illustrate the concept of fractional powers:

If the cube root of a number is squared and the result is 4, what is the number?

Finding the Unknown Number Using Fractional Powers

When faced with a problem involving fractional powers, or fraction exponents, it may seem daunting at first. However, with the correct approach and understanding of the rules, these problems can easily be solved. In this article, we will explore the process of finding an unknown number using fractional powers.

Let's say we have an unknown number, represented by y. The problem states that the cube root of y, when squared, results in 4. In mathematical terms, this can be written as (y^(1/3))^2 = 4. To simplify this equation, we can use the rule that (x^(1/n))^m = x^(m/n), resulting in y^(1/3 * 2) = 4 and y^(2/3) = 4.

To remove the fractional power, we can take the reciprocal of both sides, giving us y = 4^(3/2) = 8. Therefore, the unknown number is 8. Now let's look at some key takeaways when dealing with fractional powers.

Key Points to Remember

• Fractional powers, also known as fraction exponents, are expressions that are raised to a power that is a fraction.
• Negative fractional powers occur when the exponent is a negative fraction.
• By applying the rules of fractional powers, problems involving fractional exponents can be easily solved.
• The binomial expansion for fractional powers can be carried out by using the formula: a^(x/y) = (a^(1/y))^x.

Exploring Fractional Powers

Integrating Fractions with Powers:
When it comes to integrating expressions with fractional powers, the rules of integral calculus can be used to simplify the process.

Calculating Fractional Powers:
To calculate fractional powers, it is crucial to understand the specific rules that apply to these types of powers.

Solving Numbers to the Power of Fractions:
When solving numbers raised to a power of a fraction, the denominator becomes the root, and the numerator becomes the exponent.

Binomial Expansion with Fractional Powers:
To perform binomial expansion with fractional powers, we can use the formula of the binomial theorem.

Simplifying Algebraic Fractions with Powers:
In some cases, it may be helpful to convert algebraic fractions with powers into indices to simplify the problem before solving it.