Fourth Root
The fourth root of a number is the value that, when raised to the 4th power, equals the original number. In Elementary Algebra, you write it as √[4]x and use it as the inverse of raising a number to the fourth power.
What is the Fourth Root?
A fourth root in Elementary Algebra is the number you get when you ask, "What number to the 4th power gives this value?" If 3^4 = 81, then the fourth root of 81 is 3.
You write a fourth root with the radical symbol and a small 4 as the index: √[4]81. That index tells you this is a higher root, not a square root. Square roots have an implied index of 2, but fourth roots make the index visible, so you can tell exactly which power you are reversing.
The fourth root is the inverse of the fourth power. That means the two operations undo each other, just like subtraction undoes addition. If x = √[4]256, then x^4 = 256. This idea shows up a lot when you are solving equations that involve exponents, because you need to isolate the variable by reversing the power.
In Elementary Algebra, the cleanest way to think about a fourth root is as a power of 1/4. So √[4]x = x^(1/4). That exponent form is useful when you are combining roots with exponent rules or checking whether an answer makes sense.
One common point of confusion is that a fourth root is not the same as a square root. For example, √16 = 4, but √[4]16 = 2 because 2^4 = 16. The fourth root asks for the number multiplied by itself four times, not two times.
A quick example makes the pattern clear: √[4]81 = 3 because 3 �00 3 �00 3 �00 3 = 81. If the radicand is not a perfect fourth power, you may need to leave the answer in radical form or estimate it, depending on the problem.
Why the Fourth Root matters in Elementary Algebra
Fourth roots show up when Elementary Algebra moves beyond square roots and starts working with higher roots. Once you can read and simplify a fourth root, you can handle equations and expressions with exponents more confidently, especially when the problem is written in radical form instead of exponent form.
This term also connects directly to perfect powers. If a number is a perfect fourth power, its fourth root is a whole number, which makes simplification much easier. If it is not, you need to decide whether the expression can be simplified further or whether it should stay as a radical.
The idea matters because algebra is full of reversing operations. When a variable is raised to the 4th power, taking the fourth root is one way to isolate it. That same move shows up in problem sets, short-answer questions, and equation-solving steps where you need to show the inverse process clearly.
Fourth roots also prepare you for later root rules. Once you understand that the index tells you which power to undo, higher roots start to look patterned instead of random. That makes cube roots, fifth roots, and mixed radical expressions much easier to compare.
Keep studying Elementary Algebra Unit 9
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view galleryHow the Fourth Root connects across the course
Square Root
A square root is the 2nd root, so it asks for the number that squares to a given value. Fourth roots extend that same idea to a 4th power. Comparing the two helps you keep the index straight, since 16 = 4 but [4]16 = 2.
Higher Roots
Fourth root is one example of a higher root, which means any root with an index greater than 2. Once you know how to read the index, the method stays the same: the index tells you which power you are reversing. That pattern is the backbone of simplifying radicals in this section.
Perfect Powers
A perfect fourth power is a number you can write as some whole number raised to the 4th power. These are the easiest numbers to take fourth roots of because the answer comes out cleanly. If the radicand is not a perfect fourth power, simplification usually stops sooner.
Principal Root
The principal root is the main root value you use when a radical has a real-number answer. For fourth roots, that usually means the nonnegative root. This matters because even powers can create two possible answers in some equation settings, but the radical symbol itself points to the principal value.
Is the Fourth Root on the Elementary Algebra exam?
A quiz or problem-set question will usually ask you to evaluate, simplify, or rewrite a fourth root. You might be given a radical like [4]81 and asked for its value, or a power form like 16^(1/4) and asked to rewrite it as a radical.
You also use fourth roots when checking whether a number is a perfect fourth power. If the number is not exact, you may need to simplify by factoring out perfect fourth-power factors first. When an equation has a variable to the 4th power, taking the fourth root is one way to isolate the variable, but you still need to watch for whether the problem expects only the principal root or all possible solutions.
Key things to remember about the Fourth Root
A fourth root is the number that, when raised to the 4th power, gives the original number.
You write it with a radical and an index of 4, like [4]x.
Fourth roots are the inverse of 4th powers, so they are often used to undo exponent equations.
You can also write a fourth root as an exponent of 1/4.
A fourth root is not the same as a square root, so the index matters.
Frequently asked questions about the Fourth Root
What is Fourth Root in Elementary Algebra?
The fourth root is the number that, when multiplied by itself four times, equals the original number. In Elementary Algebra, it is written with a radical index of 4, like [4]81. It is the inverse operation of raising something to the fourth power.
How do you find the fourth root of a number?
Check whether the number is a perfect fourth power first. For example, 81 is perfect because 3^4 = 81, so [4]81 = 3. If it is not a perfect fourth power, you may simplify by factoring or leave it in radical form if the problem allows.
Is the fourth root the same as the square root?
No. A square root asks what number squared gives the original value, while a fourth root asks what number raised to the 4th power gives the original value. That difference changes the answer, even for the same number. For example, 16 = 4 but [4]16 = 2.
Can you write a fourth root as an exponent?
Yes. A fourth root is the same as raising the number to the 1/4 power. So [4]x = x^(1/4). That rewrite is handy when you are using exponent rules or checking algebraic steps.