Question

Write in simplest form. Do not use your calculator for any numerical problems. Leave your answers in radical form. $$\sqrt[3]{56}$$

Short answer

The cube root of 56 in simplest radical form is \( 2\sqrt[3]{7} \).

Step-by-step solution

Prime Factorization

Begin by finding the prime factorization of 56. The number 56 can be factored into prime numbers: 56 = 2 × 2 × 2 × 7. Using exponents, we write this as 56 = 2^3 × 7.

Rewrite Under The Radicals

Express the number inside the radical as a product of the prime factors within the cube root to obtain: \( \sqrt[3]{56} = \sqrt[3]{2^3 \times 7} \).

Simplify The Cube Root

Knowing that the cube root of a cubed number is the number itself, simplify the expression by taking the cube root of 2^3, which is 2, and leaving the 7 inside the radical since it cannot be simplified further: \( \sqrt[3]{2^3} \times \sqrt[3]{7} = 2 \sqrt[3]{7} \).

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into the product of its prime factors until all the numbers are prime. This method provides a foundational understanding of numbers that is crucial for solving various mathematical problems, including simplification of radical expressions. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.

For example, consider the number 56 from the exercise. To perform prime factorization on 56, one would systematically divide by prime numbers to find which prime factors multiply to make 56. The factors of 56 can be demonstrated as follows: 56 can be divided by 2 (which is the smallest prime number), giving us 28. Then, 28 can further be divided by 2 to give 14, and again by 2 to give 7, which is a prime number itself. Thus, the prime factorization of 56 is 2³ × 7. This breakdown is essential for the next step, which involves rewriting the expression under a radical to simplify it.

Radical Expressions

Radical expressions are expressions that contain roots. A root is broadly defined as the inverse operation of raising a number to a power. The cube root, which is involved in our example, is the specific root we are seeking when we want to find a number that, when multiplied by itself three times (cubed), gives the original number. The cube root is denoted by the radical symbol with a little 3 on the top-left side, like this: \( \sqrt[3]{} \).

In our exercise, the cube root of 56 needs to be expressed in the simplest radical form. Understanding how to handle radical expressions with prime factorization learned previously helps make simplification processes more coherent. By recognizing that numbers within a cube root can be multiplied together, we rewrite the radical expression for 56 as the product of its prime factors under the radical sign, further easing the process of simplification.

Cube Root Simplification

Cube root simplification is the act of reducing a cube root to its simplest form. The cube root of a number x is a number y that, when multiplied by itself twice more (resulting in y³), equals x. Simplifying a cube root often involves finding the largest cubed factor of a number and taking that cube root outside of the radical. The remaining factor, which is not a perfect cube, stays inside the radical.

In the step-by-step solution, once 56 was expressed as 2³ × 7, we recognized that 2³ is a perfect cube (since 2 × 2 × 2 equals 8). So, the cube root of 2³ is simply 2, taken outside of the radical. The remaining prime number 7, which is not a perfect cube, remains under the cube root symbol. This process achieves the simplification of the cube root, yielding the final expression 2 × \( \sqrt[3]{7} \) which cannot be simplified further as 7 has no further cubed factors.