Fractional Exponents

Fractional exponents are a way to represent a number that is being raised to a fractional power. In other words, it’s when we use a fraction instead of a whole number as the exponent. The number being raised to a power is called the base.

In this article, we will cover:

• 5 Fractional Exponent Rules
• How to Simplify Fractional Factors?
• How to Multiply Fractional Exponents with the Same Bases?
• How to Multiply Fractional Exponents with the Different Bases?
• Dividing Fractional Exponents
• Negative Fractional Exponents
• FAQs

In the number, say x1/y, x is the base and 1/y is the fractional exponent.

For example, in the expression 2^(3/2), 2 is the base and (3/4) is the fractional exponent. This means that 2 is being multiplied by itself 3/4 times. When we raise a number to a fractional exponent, we are taking the root of the number. So, 2^(3/2) is the same as the square root of 2 raised to the power of 3.

Fractional Exponent Rules

The rules for working with fractional exponents are similar to those for working with integer exponents, but with a few important differences.

1. Product of Fractional Exponents

When you multiply two numbers with fractional exponents, you can add the exponents.

For example, a1/m × a1/n = a(1/m + 1/n)

For example, (2^(3/4)) * (2^(1/2)) = 2^((3/4) + (1/2)) = 2^(5/4) 

2. Quotient of Fractional Exponents

When you divide two numbers with fractional exponents, you can subtract the exponents.

For example, a1/m ÷ a1/n = a(1/m – 1/n)

For example, (2^(3/4)) / (2^(1/2)) = 2^((3/4) – (1/2)) = 2^(1/4)

3. Power of a Power

When you raise a number with a fractional exponent to another power, you can multiply the exponents.

For example, (a^m)^n = a^(mn) where m and n are fractional exponents.

For example, (2^(3/4))^2 = 2^((3/4) * 2) = 2^(3/2)

4. Power of a Product

When you raise a product of numbers to a fractional exponent, you can distribute the exponent to each term.

For example, (ab)^n = (a^n)(b^n)

For example, (3*4)^(1/2) = (3^(1/2))(4^(1/2)) = (sqrt(3))*(2)

5. Power of a Quotient

When you raise a quotient of numbers to a fractional exponent, you can distribute the exponent to each term and change the sign of the exponent of the denominator.

For example, a1/m ÷ b1/m = (a÷b)1/m

For example, (3/4)^(2/3) = (3^(2/3))/(4^(2/3)) = (3^(2/3))/(4^(2/3))

It’s important to note that when working with fractional exponents, it’s necessary to be familiar with the properties of radicals and the basic operations with fractions. Also, these rules apply when the base of the fractional exponent is the same.

Simplifying Fractional Factors

Simplifying fractional exponents is the process of rewriting an expression with a fractional exponent in a simpler form. This can be done by using the properties of exponents and the rules for working with fractional exponents.

1. Using the reciprocal property

When the exponent of a number is a fraction, we can simplify it by using the reciprocal property.

For example, (a^m)^(1/n) = a^(m/n), where m and n are integers and n is not equal to zero.

For example, (2^(3/4)) = (2^(3/4))^(4/4) = 2^(3/4 * 4/4) = 2^(3/1) = 8

2. Using the properties of radicals

When the exponent of a number is a fraction, it can also be simplified by using the properties of radicals.

For example, a^(m/n) = (sqrt(a^m))^n, where m and n are integers and n is not equal to zero.

For example, (2^(3/4)) = (sqrt(2^3))^(4/4) = (sqrt(8))^(4/4) = 2^(2/1) = 4

3. Combining like terms

When an expression contains several terms with the same base and exponent, we can simplify it by combining like terms.

For example, (2^(3/4))(2^(3/4))(2^(3/4)) = (2^(3/4))^3 = 2^(3/4 * 3) = 2^(9/4)

4. Combining the above methods

we can simplify fractional exponents by combining the above methods.

For example, (2^(3/4))(2^(1/2)) = (2^(3/4))(sqrt(2^1)) = 2^(3/4 + 1/2) = 2^(5/4)

It’s important to note that when simplifying fractional exponents, it’s important to be familiar with the properties of radicals and the basic operations with fractions. Also, to simplify the expressions, it’s important to use the appropriate property or rule depending on the form of the expression.

Multiplying Fractional Exponents with Same Bases

When multiplying fractional exponents with the same base, we can add the exponents. This is because when we multiply two numbers with the same base and different exponents, we are multiplying them together multiple times.

For example, if we have the expression (a^m) * (a^n), where a is the base and m and n are the exponents, we can simplify it by adding the exponents: (a^m) * (a^n) = a^(m+n) or if we have fractional exponents then we can solve the problem as a1/m × a1/n = a(1/m + 1/n) where 1/m and 1/n are fractional exponents.

For example, (2^(3/4)) * (2^(1/2)) = 2^(3/4 + 1/2) = 2^(5/4)

It’s important to note that this rule only applies when the bases of the fractional exponents are the same. If the bases are different, we cannot add the exponents and we have to simplify the expression using other methods.

It’s also important to keep in mind that when adding the exponents, the fraction should have the same denominator, if not we have to find the common denominator before adding the exponents.

In summary, when multiplying fractional exponents with the same base, we can add the exponents to simplify the expression. This is because we are multiplying the base multiple times and this can be represented by adding the exponents.

Multiplying Fractional Exponents with Different Bases

When multiplying fractional exponents with different bases, we cannot add the exponents. This is because the bases are different and therefore the exponents cannot be combined in the same way as when the bases are the same. Instead, we have to leave the exponents in their original form and multiply the bases as normal, For example, if we have the expression (a^m) * (b^n), where a and b are the bases and m and n are the exponents, we can simplify it by multiplying the bases and keeping the exponents as they are: (a^m) * (b^n) = (ab)^(mn)

For example, (2^(3/4)) * (3^(1/2)) = (2*3)^(3/4 * 1/2) = (6)^(3/8)

It’s important to note that in this case, we can’t simplify the expression any further, and the result is the product of the bases raised to the power of the product of the exponents.

It’s also important to keep in mind that when multiplying the bases, we should be familiar with the basic operations with fractions and the properties of radicals, so we can simplify the product of the bases if needed.

In summary, when multiplying fractional exponents with different bases, we have to multiply the bases and keep the exponents as they are. This is because the bases are different and cannot be combined in the same way as when the bases are the same. The result is the product of the bases raised to the power of the product of the exponents, which can’t be simplified further.

Fractional Exponents FAQS

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