Problem 70 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. All variables represent positive real numbers. $$ \sqrt[3]{250}-4 \sqrt[3]{5}+\sqrt[3]{16} $$

Step-by-Step Solution

Answer

The simplified expression is \( 7\sqrt[3]{2} - 4\sqrt[3]{5} \).

Step 1: Simplify Each Term

Start by breaking down each radical expression into its prime factors. The expression consists of three terms: 1. \( \sqrt[3]{250} \) where 250 can be factored into \( 2 \times 5^3 \), thus \( \sqrt[3]{250} = \sqrt[3]{5^3 \times 2} = 5\sqrt[3]{2} \).2. \( -4\sqrt[3]{5} \) is already simplified.3. \( \sqrt[3]{16} \) where 16 can be expressed as \( 2^4 \), so \( \sqrt[3]{16} = \sqrt[3]{2^4} = 2\sqrt[3]{2} \).

Step 2: Combine Like Terms

Like terms in radical expressions are those that share the same radical factor. Here, both \( 5\sqrt[3]{2} \) and \( 2\sqrt[3]{2} \) are terms with \( \sqrt[3]{2} \). Add them up:\( 5\sqrt[3]{2} + 2\sqrt[3]{2} = (5 + 2)\sqrt[3]{2} = 7\sqrt[3]{2} \).

Step 3: Final Expression

Combine the result from Step 2 with the remaining term. Thus, the expression becomes:\( 7\sqrt[3]{2} - 4\sqrt[3]{5} \).Now, all terms are simplified and combined as much as possible.

Key Concepts

Understanding Prime Factors

Prime factors are the building blocks of a number, breaking it down into the smallest possible pieces that can be multiplied together to recreate the original number. To find the prime factors of a number, we start by testing it with the smallest prime number, which is 2. If the number is even, we divide it by 2 and continue dividing by 2 until it can't be divided any further. This process is repeated with the next smallest primes (3, 5, 7, etc.) until the entire number is expressed as a product of prime numbers.
For example, in the expression \( \sqrt[3]{250} \), the number 250 is broken down into prime factors: \( 250 = 5^3 \times 2 \). Recognizing prime factors allows us to break down more complex radicals in order to simplify expressions. Once a number is expressed through its prime factors, it is much easier to apply operations such as simplifying radicals.

Simplifying Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. When simplifying, you aim to "take out" any perfect roots from under the radical. A cube root, for example, simplifies any terms that can be expressed as the cube of a number.
Let's consider \( \sqrt[3]{250} \). Once we have identified its prime factors as \( 5^3 \times 2 \), we can simplify \( \sqrt[3]{5^3 \times 2} \) to \( 5\sqrt[3]{2} \). This is because \( 5^3 \) makes a perfect cube, and thus comes out of the radical, leaving behind only the non-perfect cube \( \sqrt[3]{2} \).

• For cube roots, take out terms in groups of three.
• For square roots, terms come out in pairs.

This simplification process is key to combining like terms later, as it allows you to clearly see which radicals can be grouped together.

Combining Like Terms in Radical Expressions

Combining like terms is a fundamental operation not just for simple numbers but for radicals as well. Like terms in the context of radicals share the same radical component. This allows them to be added or subtracted, just as you would with regular algebraic terms.
In our example, after simplifying the radicals, we find \( 5\sqrt[3]{2} \) and \( 2\sqrt[3]{2} \) share a common radical \( \sqrt[3]{2} \). They can therefore be combined by adding their coefficients: \( 5 + 2 = 7 \), resulting in \( 7\sqrt[3]{2} \).

• Ensure terms truly match the radical component before attempting to combine them.
• Only the coefficients outside the radicals are combined, not the numbers under the radical sign.

Practicing this will make simplifying and solving expressions with radicals more intuitive and straightforward.