Simplify Cube Root Calculator

Q: How do I simplify the cube root of 54?

To simplify the cube root of 54 , ³√(54) , follow these easy steps: Calculate the prime factorization of ³√(54) : ³√(54) = ³√(2 × 3 × 3 × 3) Write the result using powers : ³√(54) = ³√(2 × 3³) Check the exponents of the factors : 2 has exponent 1 ; 3 has exponent 3 . Since one of the exponents equals 3, we write the corresponding factor outside the root, without exponent: ³√(54) = 3 × ³√(2)

Q: What is the cube root of 1?

The cube root of 1 is 1 . 1 is the only number whose roots are always identical: starting from 0 , with 1 0 = 1 (multiplicative identity), we meet 1 1 = 1 , 1 2 = 1 , 1 3 , and so on. This implies that every possible root of 1 has result 1 .

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Meet our calculator to simplify cube roots: this handy tool will:

Clarify what a cube root is;
How to simplify the cube root: calculations step-by-step; and
How to use our simplify cube root calculator for every number you can think of.

What are you waiting for? Keep reading to find out more about this fundamental mathematical operation!

What is a cube root?

The cube root is the inverse of raising a number to the power of $3$ . In detail, the two operations are one the inverse of the other, which means that taking the cube root of a number raised to the third power returns the original number:

x = \sqrt[3]{x^3}

If the cube root of a number is an integer, we say that the number is a perfect cube. The first perfect cubes are: $1$ , $8$ , $27$ , $64$ , $125$ . Our how to find a perfect cube can teach you more about the subject!

The cube root is an essential operation in math and geometry, associating higher dimensionalities with linear quantities. The cube root of a volume, for example, is a length.

How to simplify cube root radicals

To calculate the simplified cube root, we follow a few simple steps.

Start by finding the prime factorization of the number of interest. There are multiple ways to do so: visit the prime factorization calculator to learn the math of this important step.
Group the identical factors, raising them to the proper exponent.
Check every exponent of the factors:
- If the exponent is lower than $3$ , keep the factor inside the root.
- If the exponent is equal to $3$ , bring the factor outside of the root with exponent $1$ .
- If the exponent is greater than $3$ , calculate the remainder of the division of the exponent by $3$ : this is the exponent of the factor in the root. Outside of the root, write the factor elevated to the result of the integer division between the exponent and $3$ .
Solve the powers, and multiply the factors if necessary.

We can’t calculate the simplified cube root of every number, and only some numbers are perfect cubes (three out of the first hundred positive numbers).

An example of how to simplify a cube root

Take the cube root of $283500$ . How do we simplify the cube root of this rather big number?

Find the prime factorization:

\quad \sqrt[3]{283500} = \sqrt[3]{2\!\cdot\!2\!\cdot\!3\!\cdot\!3\!\cdot\!3\!\cdot\!3\!\cdot\!5\!\cdot\!5\!\cdot\!5\!\cdot\!7}

Group the factors using powers:

\quad \sqrt[3]{283500} = \sqrt[3]{2^2\!\cdot\!3^4\!\cdot\!5^3\!\cdot\!7}

Check if there are exponents equal to $3$ . In this case, $5$ appears raised to that exponent. We carry that factor out of the root:

\quad \sqrt[3]{283500} = 5\!\cdot\!\sqrt[3]{2^2\!\cdot\!3^4\!\cdot\!7}

Check if there are exponents larger than $3$ . For example, the factor $3$ is raised to the power of $4$ . We need to split this exponent. $4=3+1$ . Every whole $3$ appearing at the exponent corresponds to multiplying the root by one time the corresponding factor. In this case, we bring out of the root $3$ a single time. The rest of the operation corresponds to the exponent of the factor inside the root:

\quad \sqrt[3]{283500} = 5\!\cdot\!3\!\cdot\!\sqrt[3]{2^2\!\cdot\!3\!\cdot\!7}

Compute the products inside and outside the root:

\quad \sqrt[3]{283500} =15\!\cdot\!\sqrt[3]{84}

That’s it. Let’s now learn how to skip all these passages and let our simplify cube root calculator do the math for us.

How to use our simplify cube root calculator

If you need to know how to simplify a cube root radical, our cube root simplifier calculator will be of great help. You can both see the result directly or calculate the cube root step-by-step simplification.

If you are interested in roots and radicals, visit our other tools like the cube root calculator, which gives you a decimal result even when you can't simplify the root, or the square root calculator.

FAQs

How do I calculate a cube root step-by-step?

The calculations to simplify a cube root of a number are not very complicated.

Start by finding the prime factorization of the number.
Group the factors using powers.
If the exponent of a factor is equal to three, bring the factor (without exponent) out of the root.
If the exponent of a factor is larger than three, bring out of the root the factor elevated to the result of the integer division exponent/3. Keep the remainder of that division as an exponent to the factor inside the root.

5 Multiply the factors to find the result.

How do I simplify the cube root of 54?

To simplify the cube root of 54, ³√(54), follow these easy steps:

Calculate the prime factorization of ³√(54):

³√(54) = ³√(2 × 3 × 3 × 3)
Write the result using powers:

³√(54) = ³√(2 × 3³)
Check the exponents of the factors:
- 2 has exponent 1;
- 3 has exponent 3.
Since one of the exponents equals 3, we write the corresponding factor outside the root, without exponent:

³√(54) = 3 × ³√(2)

What is the cube root of 1?

The cube root of 1 is 1. 1 is the only number whose roots are always identical: starting from 0, with 1⁰ = 1 (multiplicative identity), we meet 1¹ = 1, 1² = 1, 1³, and so on. This implies that every possible root of 1 has result 1.

Can you calculate the cube root of a negative number?

Yes: as opposed to square roots, defined exclusively for non-negative numbers, cube roots are defined for every possible number, negative and positive. The cube root of a negative number is a negative number, while the cube root of a positive number is a positive number. This difference stems from the fact that you need to multiply the sign an odd number of times, which allows the result to be negative in case of negative signs (as - × - × - = -).