Rational Exponents

Rational exponents are exponents of numbers that are expressed as rational numbers, that is, in a^p/q, a is the base and p/q is the rational exponent where q ≠ 0. In rational exponents, the base must be a positive integer. Rules for rational exponents are similar to the rules of integer exponents. The numerator of a rational exponent represents the power whereas the denominator of a rational exponent represents the root.

Let us explore the concept of rational exponents, how to simplify them, the relation between rational exponents and radicals along with some solved examples for a better understanding.

An exponential expression of the form a^m has a rational exponent if m is a rational number. In rational exponents, the powers and roots of a number are expressed together. Some of the examples of rational exponents are: 2^2/3, 9^5/9, 11^11/3, etc. Here the bases are positive integers and have rational exponents. Properties of general exponents also hold for the rational exponents.

Rational Exponents Definition

Rational exponents are defined as exponents that can be expressed in the form of p/q, where q ≠ 0. The general notation of rational exponents is x^m/n, where x is the base (positive number) and m/n is a rational exponent. Rational exponents can also be written as \(x^{\frac{m}{n}} = \sqrt[n]{x^m}\).

Now, let us go through some formulas of the rational exponents which are used to solve various algebraic problems. The formulas of integer exponents hold true for the rational exponents as well. Consider rational exponents with same bases a^m/n, a^p/q and a different base b^m/n

• a^m/n × a^p/q = a^{(m/n + p/q)}
• a^m/n ÷ a^p/q = a^{(m/n - p/q)}
• a^m/n × b^m/n = (ab)^m/n
• a^m/n ÷ b^m/n = (a÷b)^m/n
• a^-m/n = (1/a)^m/n
• a^0/n = a⁰ = 1
• (a^m/n)^p/q = a^{m/n × p/q}
• x^m/n = y ⇔ x = y^n/m

We can write the rational exponents expressions as radicals by identifying the powers and roots and converting them into radicals. Consider the rational exponents' expression a^m/n. Now, follow the given steps:

• Step 1: Identify the power by looking at the numerator of the rational exponent. Here in rational exponent a^m/n, m is the power.
• Step 2: Identify the root by looking at the denominator of the rational exponent. Here in rational exponent a^m/n, n is the root.
• Step 3: Write the base as the radicand, power raising to the radicand, and the root as the index. Here we can write a^m/n = ⁿ√a^m.

We can convert radicals to rational exponents as well. Consider the square root of a positive number √a. We can write the square root √a as a rational exponent. √a = a^1/2 which is a rational exponent.

Now, that we have studied the formulas of rational exponents and how to write rational exponents as radicals, let us solve some problems to learn how to simplify rational exponents. To simplify rational exponents, we need to reduce the exponential expression to its simplest form.

Example 1: Simplify the rational exponent 64^2/3

Solution: We can write 64^2/3 as 64^2/3 = (³√64)² or 64^2/3 = ³√(64)²

It is easier to determine the cube root of 64 and then squaring it as compared to finding the square of 64 and then finding its cube root. To simplify the rational exponent 64^2/3, we have

64^2/3 = (³√64)²

⇒ 64^2/3 = (4)²

⇒ 64^2/3 = 16

Hence the rational exponent 64^2/3 is simplified to 16.

Let us consider another example using the rational exponents' formulas:

Example 2: Simplify the product of rational exponents 4(2x^2/3)(7x^5/4).

Solution: To simplify the given rational exponents, we will combine the constant coefficients and separate the variables and use the formulas to simplify the rational exponents.

4(2x^2/3)(7x^5/4) = (4 × 2 × 7) × (x^2/3 × x^5/4)

= 56 (x^{2/3 + 5/4}) [Bases of the rational exponents are the same, hence we add the rational exponents]

= 56 x^23/12

Hence the product of rational exponents 4(2x^2/3)(7x^5/4) is equal to 56 x^23/12.

Exponents with fractions and decimals are referred to as non-integer rational exponents. The general format of a rational exponent is: \(a^{\frac{p}{q}}\). Here 'a' is the base and the rational number, \(\frac{p}{q} \) is the exponent. Observe the following examples of non integer rational exponents. \(2^{0.5}, 5^{\frac{2}{3}}, 11^{\frac{1}{2}}\). Consider \(27^{\frac{2}{3}}\), where 2/3 is the non integer rational exponent. It can also be written in the radical form as: ³√(27)² = (³√27)²

The non-integer rational exponents can be solved in the same way by which the exponents with integers are solved. Let us recall that if 'a' is the base and 'm' and 'n' are the exponents, which are non zero integers, the following exponent rules are used to solve the exponents.

• a^m × aⁿ = a^m+n
• a^m / aⁿ = a^m-n
• (a^m)ⁿ = a^{m × n}
• a^{- m} = 1/a^m
• \( \sqrt [n] a^m= (a^m)^{\frac{1}{n}} =a^{\frac{m}{n}}\)

Observe the following examples which show how the same exponential laws are used to solve the fractional exponents as well.

• \(\begin{align}7^{\frac{2}{3}} \times 7^{\frac{3}{2}} &= 7^{\frac{2}{3} + \frac{3}{2}}\\&=7^ {\frac{2 \times 2 + 3 \times 3}{6} } \\&= 7^{\frac{4 + 9}{6}}\\& = 7^{\frac{13}{6}}\end{align}\)
• \(\begin{align}(4^{\frac{-3}{5}})^{\frac{2}{3}} &= 4^{\frac{-3}{5} \times \frac{2}{3} } \\&= 4^{\frac{-2}{5}}\end{align}\)

Tips and Tricks on Rational Exponents

• a^m/n = ⁿ√a^m
• a^1/m × a^1/n = a^{(1/m + 1/n)}
• a^1/m ÷ a^1/n = a^{(1/m - 1/n)}
• a^1/m × b^1/m = (ab)^1/m
• a^1/m ÷ b^1/m = (a÷b)^1/m
• (a^m)^1/n = a^m/n

Related Topics

• Operations on Exponential Terms
• Negative Exponents
• Exponents Formula
• Exponential Equations
• Fractional Exponents

What are Rational Exponents in Math?

Rational exponents are exponents of numbers that are expressed as rational numbers, that is, in a^p/q, a is the base and p/q is the rational exponent where q ≠ 0.

How to Simplify Expressions with Rational Exponents?

Expressions with Rational Exponents can be simplified using formulas of the rational exponents.

What are the Properties of Rational Exponents?

The properties of rational exponents are:

• a^m/n × a^p/q = a^{(m/n + p/q)}
• a^m/n ÷ a^p/q = a^{(m/n - p/q)}
• a^m/n × b^m/n = (ab)^m/n
• a^m/n ÷ b^m/n = (a÷b)^m/n
• a^-m/n = (1/a)^m/n
• (a^m/n)^p/q = a^{m/n × p/q}

What are Radical and Rational Exponents?

Rational exponents can be expressed as radicals, that is, a^m/n = ⁿ√a^m. For example, we can write the square root of 5 as a rational exponent as well as in radical form: √5 = 5^1/2

What are Positive Rational Exponents?

Positive rational exponents are expressed with positive exponents. For example, a^1/m, where m is positive.

How to Solve Equations with Rational Exponents?

To solve equations with rational exponents, we use the rational exponents formulas such as:

• a^m/n = ⁿ√a^m
• a^1/m × a^1/n = a^{(1/m + 1/n)}
• a^1/m ÷ a^1/n = a^{(1/m - 1/n)}
• a^1/m × b^1/m = (ab)^1/m
• a^1/m ÷ b^1/m = (a÷b)^1/m
• (a^m)^1/n = a^m/n

What are the Rules of Rational Exponents?

The rules of rational exponents are:

• a^m/n × a^p/q = a^{(m/n + p/q)}
• a^m/n ÷ a^p/q = a^{(m/n - p/q)}
• a^m/n × b^m/n = (ab)^m/n
• a^m/n ÷ b^m/n = (a÷b)^m/n
• a^-m/n = (1/a)^m/n
• a^0/n = a⁰ = 1
• (a^m/n)^p/q = a^{m/n × p/q}
• x^m/n = y ⇔ x = y^n/m

What is a Non Integer Rational Exponent?

The Non Integer rational exponent is an exponent that can be represented in the form of a fraction p/q. \(a^{\frac{p}{q}}= \sqrt[q] {a^p}\). p/q can be a fraction or a decimal.

Do Non-Integer Rational Exponents Add or Multiply?

The exponents are added if the bases are multiplied. a^m × aⁿ = a^m+n whereas, the exponents are multiplied if one exponent is placed over another exponent. (a^m)ⁿ = a^{m × n} , where m and n are non-integer rational exponents.

Are Non-Integer Rational Exponents negative?

Non Integer Rational Exponents can be non positive integers,in the form of \(a^{\frac{-p}{q} }\). For example: a^-½

Where Are Negative Non-Integer Rational Exponents Used in Real Life?

In real life, negative non-integer rational exponents are used to show how small a thing is. For example, Zoologists use negative exponents to measure the different parts of bats which are extremely small to measure.