Fractional Exponents
- Introduction
- What Is the Significance of Fractional Exponents?
- Strategies for Simplifying Fractional Exponents
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
An exponent in a power term tells how many times a number (base of the power term) is multiplied by itself. When the exponent in a power term is a fractions" target="_blank" class="backlink">fraction and not a whole number, it is known as a term with a fractional exponent. Fractional exponents are a way to deal with numbers being raised to exponents that aren't whole numbers. Instead of using whole numbers like `2` or `3` as exponents, we use fractions like `1/2` or `1/3`. The number we raise the exponent to is called the base, and the fraction is the exponent. For example, in `x^{1/2}`, \( x \) is the base and `1/2` is the fractional exponent. We'll look into the rules associated with fractional exponents and learn how to work with them, including negative fractional exponents.
What Is the Significance of Fractional Exponents?
We can always associate a power term having a fractional exponent with its equivalent radical. In any exponential expression, like \( a^b \), where \( a \) represents the base and \( b \) the exponent, the introduction of fractional values for \( b \) gives rise to what we term fractional exponents. These exponents, also known as rational exponents, offer a unique approach to expressing both powers and roots simultaneously.
Let’s consider the power terms like `2^{1/2}` and `3^{2/3}` whose exponents are fractions. Within the framework of fractional exponents, `x^{a/b}` represents a power \( x^a \) over a root \( b \), where \( x \) signifies the base, \( a \) denotes the numerator, and \( b \) stands as the denominator, all being positive real numbers. Hence we can represent a power with fractional exponent using its equivalent radical as `x^{a/b} = bsqrt{x^a}`.
This helps to create a deeper understanding of mathematical expression which in turn helps in simplifying expressions with exponents.
Representation of Fractional Exponent
Some examples of fractional exponents that are widely used are given below:
Examples:
Example `1`: Convert ` \left(\frac{64}{125}\right)^{\frac{2}{3}} ` into radical form.
Solution:
`\left(\frac{64}{125}\right)^{\frac{2}{3}} = \left(\left(\frac{4^3}{5^3}\right)^{\frac{1}{3}}\right)^2 = \left(\frac{4}{5}\right)^2 = \frac{16}{25}`
Example `2`: Convert \( \sqrt[3]{8} \) into exponential form.
Solution:
`\sqrt[3]{8} = 8^{\frac{1}{3}}`
Example `3`: Convert ` \left(\frac{27}{8}\right)^{\frac{2}{3}} ` into radical form.
Solution:
`\left(\frac{27}{8}\right)^{\frac{2}{3}} = \left(\left(\frac{3^3}{2^3}\right)^{\frac{1}{3}}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}`
Example `4`: Convert \( \sqrt{15} \) into exponential form.
Solution:
\( \sqrt{15} = 15^{\frac{1}{2}} \)
Example `5`: Convert ` \left(\frac{343}{512}\right)^{\frac{2}{3}} ` into radical form.
Solution:
`\left(\frac{343}{512}\right)^{\frac{2}{3}} = \left(\left(\frac{7^3}{8^3}\right)^{\frac{1}{3}}\right)^2 = \left(\frac{7}{8}\right)^2 = \frac{49}{64}`
Strategies for Simplifying Fractional Exponents
We use various exponent rules to simplify exponential expressions. Likewise, we can use these rules to simplify expressions with fractional exponents too.
We're simplifying fractions with exponents by combining two basic methods: multiplying and dividing. This means making expressions or exponents simpler and easier to understand.
- Multiplication of fractional exponents
Multiplication can be categorized into two scenarios.
Firstly, when dealing with the same base, we add the exponents and express the result using the common base using the formula `a^{1/m} \times a^{1/n} = a^{(1/m + 1/n)}`.
For instance, when multiplying ` 3^{1/3}` and `3^{3/4}`, the sum of exponents is `\frac{1}{3} + \frac{3}{4} = \frac{13}{12}`, resulting in `3^{13/12}`.
Secondly, when the powers are the same but the bases differ, we simply multiply the bases and retain the exponent as common, resulting in `a^{1/m} \times b^{1/m} = (a \times b)^{1/m}`.
For instance, `15^{3/4} \times 3^{3/4}` simplifies to `(15\times3)^{3/4} = 45^{3/4}`.
Example `1`: Simplify `x^{1/2} \times x^{1/3}`
Solution:
Here, we're multiplying \(x\) raised to the power of `1/2` by \(x\) raised to the power of `1/3`. To simplify, we add the exponents together:
`x^{1/2} \times x^{1/3} = x^{1/2 + 1/3} = x^{5/6}`
Example `2`: Simplify `a^{2/3} \times b^{2/3}`
Solution:
We're multiplying \(a\) raised to the power of `2/3` by \(b\) raised to the power of `2/3`. Here, the exponents `2/3` are the same, but the bases \(a\) and \(b\) are different.
Hence, `a^{2/3} \times b^{2/3} = (a \times b)^{2/3} = (ab)^{2/3}`
- Division of fractional exponents
Like multiplication, division can be categorized into two scenarios.
Firstly, when dealing with different powers but the same bases, we subtract the exponents to express it as `a^{1/m} \div a^{1/n} = a^{(1/m - 1/n)}`.
For instance, `2^{3/4} \div 2^{1/2}` yields `2^{3/4 - 1/2} = 2^{1/4}`.
Secondly, when the powers are the same but the bases differ, we simply divide the bases and retain the exponent as common, resulting in `a^{1/m} \div b^{1/m} = (a \div b)^{1/m}`.
For instance, `27^{3/5} \div 3^{3/5}` simplifies to `(27/3)^{3/5} = 9^{3/5}`.
Example `1`: Simplify `\frac{x^{1/3}}{x^{1/6}}`
Solution:
`\frac{x^{1/3}}{x^{1/6}} = x^{(1/3) - (1/6)} = x^{(2/6) - (1/6)} = x^{1/6}`.
Therefore, `\frac{x^{1/3}}{x^{1/6}} = x^{1/6}`.
Example `2`: Simplify `\frac{15^{3/4}}{3^{3/4}}`
Solution:
In this example, the exponents `3/4` are the same, but the bases \(15\) and \(3\) are different. We can simplify this expression by dividing the bases, keeping the exponent the same.
`\frac{15^{3/4}}{3^{3/4}} = 5^{3/4}`
So, the result is `5^{3/4}`, where the exponents are the same but the bases are different.
- Negative fractional exponents or rational exponents
This happens when powers carry a negative sign alongside fractional values. To solve such expressions, we apply the principle `a^{-m} = \frac{1}{a^m}` initially, followed by further simplification using the general formula ` a^{-m/n} = (\frac{1}{a})^{m/n}`. For instance, simplifying `343^{-1/3}` involves raising the reciprocal of the base, `\frac{1}{343}`, to the power without the negative sign, resulting in `\frac{1}{7}`.
Example `1`: `x^{-1/2}`
Solution:
This expression represents the reciprocal of the square root of `x`. For instance, if \( x = 4 \), then `x^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2}`.
Example `2`: `\frac{1}{y^{-3/4}}`
Solution:
This expression represents the fourth root of `y` cube in the denominator. For example, if \( y = 16 \), then `\frac{1}{y^{-3/4}} =`\( {\sqrt[4]{16}^3} = 2^3 = 8\).
Understanding the process and the rules involved in simplifying fractional exponents is important. This helps us to tackle complex exponential expressions with ease. Please note that it comes very handy to know the perfect square and perfect cubes at least for numbers from `1` to `15`. It helps you to comprehend and rewrite the bigger numbers in their square form and /or cube form making the process more faster and simpler.
Solved Examples
Example `1`: Simplify `16^{1/4}`.
Solution:
`16` can be expressed as \(2 \times 2 \times 2 \times 2\), or \(2^4\).
So, we have `16^{1/4} = (2^4)^{1/4}`.
Applying the power of a power rule of exponents,
`(2^4)^{1/4} = 2^{4/4} = 2^1 = 2`
So `16^{1/4}` in its simplest form can be written as `2`.
Example `2`: Simplify `5^{2/3} \times 5^{3/4}`.
Solution:
Since both exponents have the same base, we can add the exponents.
Sum of exponents `= \frac{2}{3} + \frac{3}{4} = \frac{8+9}{12} = \frac{17}{12}`
Therefore, `5^{2/3} \times 5^{3/4} = 5^{17/12}`.
Example `3`: Simplify `\frac{81^{3/4}}{9^{1/2}}`.
Solution: To simplify `\frac{81^{3/4}}{9^{1/2}}`, let's express both terms with the same base:
`1`. `81^{3/4}`: We can rewrite \( 81 \) as `3^4` since `3^4 = 81`. So, `81^{3/4} = (3^4)^{3/4} = 3^{4 \times \frac{3}{4}} = 3^3 = 27`.
`2`. `9^{1/2}`: \( 9 \) can be written as \( 3^2 \) since \( 3^2 = 9 \). So, `9^{1/2} = (3^2)^{1/2} = 3^{2 \times \frac{1}{2}} = 3^1 = 3`.
Now, we can simplify the expression:
`\frac{81^{3/4}}{9^{1/2}} = \frac{27}{3} = 9`
So, `\frac{81^{3/4}}{9^{1/2}}` simplifies to \( 9 \).
Example `4`: Let's evaluate `\frac{18^{1/2}}{2^{1/2}}`.
Solution: In this problem, both exponents are the same, but the bases are different. So, we can simplify this as `(\frac{18}{2})^{1/2} = 9^{1/2} = 3`. Therefore, the answer is `3`.
Example `5`: Simplify `2\times3^{3/2}`.
Solution: We can rewrite `3^{3/2}` as `\sqrt{3^3}`.
So, `2\sqrt{3}^{3/2} = 2(\sqrt{3^3}) = 2 \times 3\sqrt{3} = 6\sqrt{3}`
Example `6`: Evaluate `5^{\frac{4}{3}}`.
Solution: We can rewrite \( {5}^{\frac{4}{3}} \) as \( (\sqrt[3]{5})^4 \).
\( {5}^{\frac{4}{3}} = (\sqrt[3]{5})^4 = \sqrt[3]{5^4} = \sqrt[3]{625} \)
Example `7`: Simplify \( \sqrt{2} \times \sqrt[4]{2} \).
Solution: We can rewrite \( \sqrt{2} \) as `2^{\frac{1}{2}}` and \( \sqrt[4]{2} \) as `2^{\frac{1}{4}}`
So, \( \sqrt{2} \times \sqrt[4]{2}\)`= 2^{\frac{1}{2}} \times 2^{\frac{1}{4}} = 2^{\frac{1}{2} + \frac{1}{4}} = 2^{\frac{3}{4}} =` \(\sqrt[4]{2^3} = \sqrt[4]{8}\)
Practice Problems
Q`1`. Simplify `2^{\frac{3}{2}}`.
- \( \sqrt{2} \)
- \( 2\sqrt{2} \)
- \( \sqrt{8} \)
- \( \frac{1}{4} \)
Answer: c
Q`2`. Simplify `4^{-\frac{1}{2}}`.
- \( \frac{1}{4} \)
- \( \frac{1}{2} \)
- \( 2 \)
- \( \frac{1}{2} \)
Answer: d
Q`3`. Simplify `9^{\frac{3}{2}}`.
- \( 3 \)
- \( 27 \)
- \( 81 \)
- \( 6 \)
Answer: b
Q`4`. Simplify `\left(\frac{1}{216}\right)^{-\frac{2}{3}}`.
- \( \frac{1}{36} \)
- \( \frac{1}{2} \)
- \( 64 \)
- \( 36 \)
Answer: d
Q`5`. Evaluate `16^{\frac{-3}{4}}`.
- \( \frac{1}{64} \)
- \( \frac{1}{8} \)
- \( \frac{1}{16} \)
- \( \frac{1}{256} \)
Answer: b
Q`6`. Simplify the expression \( \sqrt[3]{x^{\frac{5}{2}}} \). Which of the following is equivalent to the given expression?
- \( x^{\frac{5}{6}} \)
- \( x^{\frac{5}{3}} \)
- \( x^{\frac{5}{4}} \)
- \( x^{\frac{15}{2}} \)
Answer: a
Q`7`. Evaluate `(16)^{\frac{3}{4}}`. Which of the following represents the result in radical form?
- \( \sqrt[4]{16^3} \)
- \( \sqrt[3]{16^4} \)
- \( \sqrt{16^3} \)
- \( \sqrt[4]{16}^3 \)
Answer: a
Q`8`. If \( \sqrt[5]{x^3} = 2 \), what is the value of \( x \)?
- \( x = 2^5 \)
- \( x = 2^3 \)
- \( x = 2^{\frac{3}{5}} \)
- \( x = 2^{\frac{5}{3}} \)
Answer: d
Frequently Asked Questions
Q`1`. What are the rules for simplifying fractional exponents?
Answer: To simplify fractional exponents, we apply the laws of exponents or exponent rules. These rules are outlined as follows:
Rule `1`: `a^{\frac{1}{m}} \times a^{\frac{1}{n}} = a^{\left(\frac{1}{m} + \frac{1}{n}\right)}`
Rule `2`: `a^{\frac{1}{m}} \div a^{\frac{1}{n}} = a^{\left(\frac{1}{m} - \frac{1}{n}\right)}`
Rule `3`: `a^{\frac{1}{m}} \times b^{\frac{1}{m}} = (ab)^{\frac{1}{m}}`
Rule `4`: `a^{\frac{1}{m}} \div b^{\frac{1}{m}} = \left(\frac{a}{b}\right)^{\frac{1}{m}}`
Rule `5`: `a^{-\frac{m}{n}} = \left(\frac{1}{a}\right)^{\frac{m}{n}}`
Q`2`. What is the process for combining power terms with fractional exponents?
Answer: There is no specific rule for combining power terms with fractional exponents. Their combination relies on simplifying the powers if feasible. For instance, `9^{1/2} + 125^{1/3} = 3 + 5 = 8`.
Q`3`. How should negative fractional exponents be simplified?
Answer: When a negative exponent is encountered, we negate the sign from the exponent and find the reciprocal of the base. For instance, `8^{-1/2}` equals `\left(\frac{1}{8}\right)^{1/2}`.
Q`4`. What strategies are employed to simplify expressions containing fractional exponents?
Answer: To simplify, utilize rules such as combining exponents when bases match, or convert fractional exponents into radical form and perform regular operations.
Q`5`. What method is used for dividing fractional exponents?
Answer: A division involving fractional exponents having identical bases requires subtracting the exponents. On the other hand, division featuring different bases but identical powers involves dividing the bases with the exponents remaining the same.