Cube Roots

We know that [latex]5^2=25, \text{ and }\sqrt{25}=5[/latex], but what if we want to “undo” [latex]5^3=125, \text{ or }5^4=625[/latex]? We can use higher order roots to answer these questions.

While square roots are probably the most common radical, we can also find the third root, the fifth root, the [latex]10th[/latex] root, or really any other nth root of a number. Just as the square root is a number that, when squared, gives the radicand, the cube root is a number that, when cubed, gives the radicand.

The cube root of a number is written with a small number  [latex]3[/latex], called the index, just above the v-part radical symbol. It looks like [latex]\sqrt[3\;]{{\;}}[/latex]. This little [latex]3[/latex] distinguishes cube roots from square roots. For a square root the index is [latex]2[/latex] but it is such a common root that the index is not written.

Suppose we know that [latex]{a number}^{3}=8[/latex]. We want to find what number raised to the [latex]3[/latex]rd power is equal to [latex]8[/latex]. Since [latex]{2}^{3}=8[/latex], we say that [latex]2[/latex] is the cube root of [latex]8[/latex]. In the next example, we will evaluate the cube roots of some perfect cubes.

As we saw in the last example, there is one interesting fact about cube roots that is not true of square roots. Negative numbers cannot have real number square roots, but negative numbers can have real number cube roots! What is the cube root of [latex]−8[/latex]? [latex]\sqrt[3\;]{-8}=-2[/latex] because [latex]-2\cdot -2\cdot -2=-8[/latex]. Remember, when we are multiplying an odd number of negative numbers, the result is negative! Consider [latex]\sqrt[3\;]{{{(-1)}^{3}}}=-1[/latex].

We can also use factoring to simplify cube roots such as [latex]\sqrt[3\;]{125}[/latex]. We read this as “the third root of [latex]125[/latex]” or “the cube root of [latex]125[/latex].” To simplify this expression, look for a number that, when multiplied three times , equals [latex]125[/latex]. Let’s factor [latex]125[/latex] and find that number.

The prime factors of [latex]125[/latex] are [latex]5\cdot5\cdot5[/latex], which can be rewritten as [latex]5^{3}[/latex]. The cube root of a cubed number is the number itself, so [latex]\sqrt[3\;]{{{5}^{3}}}=5[/latex]. We have found the cube root, the three identical factors that when multiplied together give [latex]125[/latex]. [latex]125[/latex] is known as a perfect cube because its cube root is an integer.

nth Roots

We learned above that the cube root of a number is written with a small number [latex]3[/latex], which looks like [latex]\sqrt[3\;]{a}[/latex]. This number placed just outside and above the radical symbol and is called the index.

We can apply the same idea to any exponent and its corresponding root. The nth root of [latex]a[/latex] is a number that, when raised to the nth power, gives [latex]a[/latex]. For example, [latex]3[/latex] is the 5th root of [latex]243[/latex] because [latex]{\left(3\right)}^{5}=243[/latex]. If [latex]a[/latex] is a real number with at least one nth root, then the principal nth root of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that, when raised to the nth power, equals [latex]a[/latex].

The principal nth root of [latex]a[/latex] is written as [latex]\sqrt[n\;]{a}[/latex], where [latex]n[/latex] is a positive integer greater than or equal to [latex]2[/latex]. In the radical expression, [latex]n[/latex] is called the index of the radical.

The nth roots of a real number are categorized into two groups: odd roots when the index is an odd number, and even roots when the index is an even number. Square roots are even roots, since 2 is an even number. The square root of a negative real number is undefined in the set of real numbers. The same can be said for all even roots: the even root of a negative real number is undefined in the set of real numbers. This is because any real number raised to an even power is always positive. For example, [latex](-3)^4=81[/latex]; [latex](-2)^6=64[/latex]; [latex](-1)^126=1[/latex].

However, odd roots, like the cube root, exist for all real numbers. For example, [latex](-2)^5=-32[/latex], so [latex]\sqrt[5\;]{-32}=-2[/latex], and since [latex]4^3=64[/latex], [latex]\sqrt[3\;]{64}=4[/latex]. Odd roots take on the sign of the radicand.

The steps to consider when simplifying a radical are outlined below.

The video shows more examples of how to evaluate nth roots.

We can find the odd root of a negative number, but we cannot find the even root of a negative number in the set of real numbers. This means we can evaluate the radicals [latex]\sqrt[3\;]{-81},\ \sqrt[5\;]{-64}[/latex], and [latex]\sqrt[7\;]{-2187}[/latex] because they all have an odd numbered index, but we cannot evaluate the radicals  [latex]\sqrt[{}]{-100},\ \sqrt[4\;]{-16}[/latex], or [latex]\sqrt[6\;]{-2,500}[/latex] because they all have an even numbered index.

Rational Exponents

Square roots are most often written using a radical sign, like this, [latex]\sqrt{4}[/latex]. But there is another way to represent them. We can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, [latex]\sqrt{4}[/latex] can be written as [latex]{{4}^{\small\frac{1}{2}}}[/latex].

Radicals and rational exponents are alternate ways of expressing the same thing.  In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.

Radical Form	Exponent Form	Principal Root
[latex]\sqrt{16}[/latex]	[latex]{{16}^{\tfrac{1}{2}}}[/latex]	[latex]4[/latex]
[latex]\sqrt{25}[/latex]	[latex]{{25}^{\tfrac{1}{2}}}[/latex]	[latex]5[/latex]
[latex]\sqrt{100}[/latex]	[latex]{{100}^{\tfrac{1}{2}}}[/latex]	[latex]10[/latex]

Use the example below to familiarize yourself with the different ways to write square roots.

In the following video, we show another example of filling in a table to connect the different notation used for roots.

Let us look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number [latex]3[/latex].

Radical Form	Exponent Form	Principal Root
[latex]\sqrt[3\;]{8}[/latex]	[latex]{{8}^{\tfrac{1}{3}}}[/latex]	[latex]2[/latex]
[latex]\sqrt[3\;]{125}[/latex]	[latex]{{125}^{\tfrac{1}{3}}}[/latex]	[latex]5[/latex]
[latex]\sqrt[3\;]{1000}[/latex]	[latex]{{1000}^{\tfrac{1}{3}}}[/latex]	[latex]10[/latex]

These examples help us model a relationship between radicals and rational exponents: namely, that the nth root of a number can be written as either [latex]\sqrt[\;n]{x}[/latex] or [latex]{{x}^{\frac{1}{n}}}[/latex].

Radical Form	Exponent Form
[latex]\sqrt{x}[/latex]	[latex]{{x}^{\tfrac{1}{2}}}[/latex]
[latex]\sqrt[3\;]{x}[/latex]	[latex]{{x}^{\tfrac{1}{3}}}[/latex]
[latex]\sqrt[4\;]{x}[/latex]	[latex]{{x}^{\tfrac{1}{4}}}[/latex]
…	…
[latex]\sqrt[n\;]{x}[/latex]	[latex]{{x}^{\tfrac{1}{n}}}[/latex]

In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of [latex]\frac{1}{2}[/latex] translates to the square root, an exponent of [latex]\frac{1}{5}[/latex] translates to the fifth root or [latex]\sqrt[5\;]{x}[/latex], and [latex]\frac{1}{8}[/latex] translates to the eighth root or [latex]\sqrt[8\;]{x}[/latex].

When converting from radical to rational exponent notation, the index of the root becomes the denominator of the exponent. If we start with a square root, we will have an exponent of [latex]\frac{1}{2}[/latex] on the radicand. On the other hand, if we start with an exponent of [latex]\frac{1}{3}[/latex] we will use a cube root. The following statement summarizes this idea.

Simplifying Nth Roots on a Calculator

In the same way we can find a square root on a calculator, we can find a higher root in the calculator to estimate what the root will be. For example, consider the [latex]\sqrt[\;5\;]{250}[/latex]. The calculator, depending on your type of calculator, has two different function keys that will take nth roots. The first is the key [latex]\sqrt[y\;]{x}[/latex], the second is converting the root to a rational exponent and using the exponent key on the calculator.

On the free online scientific calculator from Desmos, we find [latex]\sqrt[7\;]{128}[/latex] by clicking on the nth root button, followed by the index 7, followed by the radicand, 150 after clicking under the radical in the display. The answer shows on the right side of the display and rounds to three decimal places as 2.046.

We could get the same answer by writing [latex]\sqrt[7\;]{128}[/latex] as [latex]128^\frac{1}{7}[/latex] and clicking 128 then the exponent button [latex]a^b[/latex] followed by the exponent [latex]1\div 7[/latex]. As we can see, we get the same answer.