Question

Addition and Subtraction of Radicals. Combine as indicated and simplify. $$\sqrt[3]{384}-\sqrt[3]{162}+\sqrt[3]{750}$$

Short answer

The simplified form of the expression is \(6\sqrt[3]{6}\).

Step-by-step solution

Factor Into Prime Factors

Begin by factoring each number within the cube roots into its prime factors: - For 384, find that it is equal to \(2^7 * 3\), - For 162, find that it is equal to \(2 * 3^4\), - For 750, find that it is equal to \(2 * 3 * 5^2 * 5\).

Group the Prime Factors into Cubes

Group the prime factors into triples that can form cubes: - For \(2^7 * 3\), group as \(2^3 * 2^3 * 2 * 3\) (where \(2^3\) is a cube), - For \(2 * 3^4\), group as \(3^3 * 2 * 3\) (where \(3^3\) is a cube), - For \(2 * 3 * 5^2 * 5\), group as \(5^3 * 2 * 3\) (where \(5^3\) is a cube).

Simplify the Cube Root of Each Term

Take the cube root of each grouped term. This gives:- For \(\sqrt[3]{2^3 * 2^3 * 2 * 3}\), this simplifies to \(2 * 2 * \sqrt[3]{2 * 3} = 4\sqrt[3]{6}\),- For \(\sqrt[3]{3^3 * 2 * 3}\), this simplifies to \(3 * \sqrt[3]{2 * 3} = 3\sqrt[3]{6}\),- For \(\sqrt[3]{5^3 * 2 * 3}\), this simplifies to \(5 * \sqrt[3]{2 * 3} = 5\sqrt[3]{6}\).

Combine Like Terms

Combine like terms that have the same radical part: \(4\sqrt[3]{6} - 3\sqrt[3]{6} + 5\sqrt[3]{6} = (4 - 3 + 5)\sqrt[3]{6} = 6\sqrt[3]{6}\).

Key concepts

Simplifying Cube Roots

When you encounter a cube root, you're looking for a number that, when multiplied by itself three times, gives you the original number under the radical. Simplifying cube roots involves breaking down the number inside the cube root into its prime factors and then grouping them in sets of three, which helps in pulling them out of the radical sign. For example, to simplify \(\sqrt[3]{8}\), you would note that 8 is \(2^3\), so the cube root of 8 is simply 2.
In the exercise \(\sqrt[3]{384}-\sqrt[3]{162}+\sqrt[3]{750}\), we begin by expressing each number as a product of its prime factors. This is crucial because it allows us to identify sets of three identical factors, making them perfect cubes. A perfect cube within a cube root can be simplified by taking the cube root of that set, leaving us with an integer rather than a radical. For example, in \(\sqrt[3]{2^7 \cdot 3}\), by recognizing \(2^3\) as a cube, we can simplify it directly to 2, which exits the radical sign, significantly simplifying our expression.

Prime Factorization

Prime factorization is the process of decomposing a number into a product of prime numbers. This is fundamental when simplifying radicals because it makes it possible to identify and extract cubes easily. A prime number is a number greater than 1 that has no divisors other than 1 and itself. The numbers 2, 3, 5, 7, 11, and so forth are prime numbers.
To factor a number into its primes, divide it by the smallest prime number possible and continue dividing the quotient by prime numbers until you are left with a product of primes. In the context of our exercise, the number 384 is factored into \(2^7 \cdot 3\), where both 2 and 3 are prime. This process is not just a step towards simplifying cube roots; it’s a foundational skill in algebra that helps in various aspects of problem-solving. Recognizing the role of prime factorization in breaking down complex expressions provides students with a powerful tool for tackling a wide range of mathematical challenges.

Combining Like Terms

Once we've simplified radicals, we can add or subtract them if they share the same radical component, which means they have the same index and radicand (the number under the radical sign). This process is known as combining like terms. It's much like combining similar variables; for instance, \(2x + 3x\) simplifies to \(5x\) because the variable part (\(x\)) is the same in both terms.
In our example, after simplifying, we were left with terms that all contained \(\sqrt[3]{6}\). Since the cube root of 6 is the common radical part, we can combine these terms just like we would with similar variables. The coefficients (the numbers in front of the radicals) are then added or subtracted accordingly. \(4\sqrt[3]{6} - 3\sqrt[3]{6} + 5\sqrt[3]{6}\) simplifies to \(6\sqrt[3]{6}\) because when you combine the coefficients (4, -3, and 5), they sum up to 6 which is then multiplied by the shared radical part. Recognizing and combining like terms is essential for simplifying expressions and solving equations effectively.