In this explainer, we will learn how to perform operations and simplifications with expressions that involve rational exponents.
Exponents have many real-world applications—for example, in scientific scales such as the pH or Richter scale, in physics with the inverse square law of electromagnetism, gravity, or the half-life of radioactive material, in engineering when taking measurements and calculating multidimensional quantities, in computing when describing the capacity of memory such as RAM or ROM, in finance with compound interest, or in biology when describing the growth or spread of bacteria or viruses, to name a few.
A rational exponent is one where the exponent is a rational number (i.e., an integer or the quotient of two integers).
Let’s first recall exponents for integer powers. For positive integers, we have the following definition.
Definition: Positive Integer Exponents
The general form of a base raised to the power of , where is a positive integer, is given by where there are factors of the base (i.e., is multiplied by itself repeatedly and it appears in the product times).
For example, is the square of the number 3 and is the cube of the number 3, and so on.
For negative integer powers, we just take the reciprocal of the positive exponent.
Definition: Negative Integer Exponents
The general form of a base raised to the power of , where is a positive integer, is given by
For example, .
Recall that, to find the product of two powers that have the same base, we have the product rule for exponents,
In other words, if the bases are the same when we multiply, we can add the exponents. For example, , which is expected from the definition as 4 appears in the product twice for and thrice for , making a total of 5 times in the product. We also have a rule for the product of two different bases and raised to the same power, in particular
In other words, if the exponents are the same then we can multiply the bases first and then evaluate the exponent of the result. For example, , which is expected since . Another way of thinking about this is that exponentiation distributes over multiplication for integer exponents.
We also have a rule where we can raise to another power as
In other words, raising a number with base and exponent to another exponent is the same as raising to . For example, , which is expected since .
The same rules, such as the product of a base with different exponents or the product of two different bases with the same exponent, apply for negative exponents. This is because we have a positive exponent in the reciprocal.
We can see this directly by the definition. For example, if we have the same base with different negative exponents and , where and are positive, we can write this as
Similarly, if we divide two different powers of the same base , we have
For example, . We also note that, for zero exponents and for any nonzero base , we have . We can see this directly from the rules above by taking the product of raised to and :
On the other hand, from the definition of the negative exponent,
Thus, we have for any nonzero base , as expected. We can also take the product of two different bases and raised to the same negative power:
This also means that when we raise a fraction to an integer power , we can express this as
For example, .
When simplifying rational exponents with different bases, it is convenient to try to write each power with the same exponent using . For example, can be written as .
We can also simplify exponents by writing their bases in terms of their prime factorization, if the base is an integer. This allows us to split the expression with like bases and apply the product rule with .
Let’s consider an example where we use the laws of exponents with a negative exponent to simplify a rational algebraic expression.
Example 1: Simplifying Rational Algebraic Expressions Using Laws of Exponents with Negative Exponents
Simplify .
Answer
In this example, we will simplify a rational algebraic expression using the following laws of exponents:
Since the exponent appears a lot in the given expression, we will attempt to write each factor in terms of this exponent alone. The factors that appear can be rewritten as
We can also decompose the bases into their prime factorization:
Therefore, we have
Thus, using these, we can write the expression as
Let’s consider another example, with slightly different exponents that can be simplified using the laws of exponents.
Example 2: Simplifying Rational Algebraic Expressions Using Laws of Exponents
Simplify .
Answer
In this example, we will simplify a rational algebraic expression using the following laws of exponents:
Since the exponent appears a lot in the given expression, we will attempt to write each factor in terms of this exponent alone. The factors that appear can be rewritten as
We can also decompose the bases into their prime factorization:
Therefore, the expression can be written as
So far, everything has been considered for integers. But what if we have fractional exponents? Let’s first recall the root of a number .
An root of a real number , where is a positive integer, is a real number that when raised to the power of , gives :
If and is an even number, then no such real root exists. Otherwise, we denote the positive root as , for all positive integer values of .
If is even, then we have another real root given by , as this number raised to the power of also gives : since when is even.
Thus, if is even and , the real roots are . If is odd, we always have one unique real root, .
For example, the square roots of 9 are 3 and since , but the square root of does not exist. Also, the cube root of 27 is 3 and the cube root of is .
Definition: The 𝑛th Root and 1/𝑛 Exponent
A fractional exponent , where is an integer, of a number can be expressed in terms of an root of a number as
For example, .
Let’s consider an example where we use this to simplify an expression.
Example 3: Simplifying an Expression Raised to a Rational Exponent
Simplify , where and are positive constants.
Answer
In this example, we will simplify an expression raised to a rational exponent.
We will make use of the law of exponents, which states
First, note that
Therefore, the expression can be written as
Now, let’s consider an example where we simplify an expression raised to a rational exponent.
Example 4: Simplifying an Expression Raised to a Rational Exponent
Expand , where is a real constant.
Answer
In this example, we will simplify an expression raised to a rational exponent.
We will make use of the power law, and the fact that for any rational exponent .
Therefore, the expression can be written as
This is option B.
Again, the same rules apply for the exponents that we have established for integers, and we can use these to write a number raised to any rational power. A rational number can be expressed as , where and are integers and is nonzero.
Definition: Fractional Exponent
A fractional exponent , where and are integers and , of a number can be expressed as assuming a real root exists.
We can see this from the rule and the root of a number (assuming a real root exists) as or, equivalently,
In order to simplify numerical expressions with different bases and rational exponents, it is convenient to write them in terms of their prime factorization, if the base is an integer. This allows us to split the expression with like bases and apply the product rule for exponents with .
Let’s consider an example where we have to do this to simplify an expression.
Example 5: Simplifying an Expression Involving Integers Raised to Rational Exponents
Simplify .
Answer
In this example, we will simplify an expression involving integers raised to rational exponents. We will make use of the following laws of exponents:
We can also decompose the bases into their prime factorization:
Applying the appropriate power to each of these bases, we have
Thus, using these, the given expression can be written as
If the base is a decimal or fraction, then we should write the base as a fraction and then decompose the numerator and denominator into their prime factors. Now, let’s look at a few examples of this.
Example 6: Simplifying a Numerical Expression Raised to Rational Exponents
Simplify .
Answer
In this example, we will simplify a numerical expression raised to rational exponents. We will make use of the following laws of exponents:
Let’s first write each base as a fraction and express the denominator and numerator in terms of their prime factorization:
Applying the appropriate power to each of these bases, we have
Thus, using these, the expression can be written as
Example 7: Simplifying a Numerical Expression Raised to Rational Exponents
Simplify .
Answer
In this example, we will simplify a numerical expression raised to rational exponents. We will make use of the following laws of exponents:
Let’s first write each base as a fraction and then express the denominator and numerator in terms of their prime factorization:
Applying the appropriate power to each of these bases, we have
Thus, using this, the expression can be written as
Finally, let’s consider an example where we have to simplify an algebraic expression involving rational and negative exponents.
Example 8: Simplifying Algebraic Expressions Using Laws of Exponents Involving Rational and Negative Exponents
Determine the simplest form of .
Answer
In this example, we will simplify an algebraic expression involving rational and negative exponents. We will make use of the laws of exponents along with the fact that for any rational number . Let’s assume that is a rational number and begin by separating the exponents in each of the factors:
The prime factorization of each of the bases is
Applying the appropriate powers to the bases, we have
Thus, the expression can be written as
Key Points
- The laws of exponents for bases and
and all rational exponents
and are as follows:
Product rules Negative exponent Quotient rules Power rules - We also have for any nonzero number and for any rational number .
- When simplifying numerical expressions with different bases or exponents, we should try to rewrite the expression in such a way that we have common bases or exponents. We can do this by using the laws of exponents and by prime factorization of the denominator and numerator of the different bases.
