Problem 23 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ -5 \sqrt[3]{625}+\sqrt[3]{40} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-23 \sqrt[3]{5}\).

Step 1: Simplify the Cube Roots

First, we need to simplify each of the cube roots individually. Start with \( \sqrt[3]{625} \). Notice that 625 can be broken down into its prime factors: \( 625 = 5 \times 5 \times 5 \times 5 = 5^4 \). The cube root of 625 can be rewritten as \( \sqrt[3]{5^4} = 5 \times \sqrt[3]{5} \). For \( \sqrt[3]{40} \), we write 40 as \( 2^3 \times 5 \), so \( \sqrt[3]{40} = \sqrt[3]{2^3 \times 5} = 2 \times \sqrt[3]{5} \).

Step 2: Combine Like Terms

Now that we have the simplified forms of the cube roots, substitute them back into the original expression: \( -5 \times 5 \times \sqrt[3]{5} + 2 \times \sqrt[3]{5} \). This simplifies to \( -25 \sqrt[3]{5} + 2 \sqrt[3]{5} \).

Step 3: Factor and Simplify the Expression

Since both terms have \( \sqrt[3]{5} \) as a common factor, we can factor it out: \( (-25 + 2) \times \sqrt[3]{5} \). Simplifying the terms in the parenthesis gives \( -23 \sqrt[3]{5} \).

Key Concepts

Simplifying Radical Expressions

When tackling radical expressions, simplifying them is often the first step. In our exercise, we dealt with cube roots, which are a type of radical expression. Simplifying these involves expressing these roots in their most reduced form. By doing this, calculations become straightforward.

To simplify a cube root, you should look at the number inside the root and think about its prime factors. For example, taking the cube root of 625 involves breaking it down to prime factors, which are powers of smaller numbers. Once this is done,

• Identify any perfect cubes (in our case, numbers raised to a power that is a multiple of three).
• Take out those as whole numbers outside the radical.

This practice makes complex expressions much more manageable. It's like untangling a knot, where you gradually simplify it until you have a neat and clean result.

Prime Factorization

Prime factorization is a crucial step in simplifying radicals. It involves breaking down a number into its simplest building blocks, which are prime numbers. Let's look closer at how we can utilize this method.

Consider the number 625 as in our exercise. Using prime factorization, we break it down into

• 625 = 5 × 5 × 5 × 5 (or 5 raised to the power of 4).

Now, for cube roots, we identify groups of three. Since a cube root is essentially unsolved until all groups of three primes have been taken out, this method lets us easily solve or simplify such expressions. It works wonders also for numbers like 40, which further breaks down to:

• 40 = 2 × 2 × 2 × 5 (or 2 raised to the power of 3 times 5).

Recognizing these groups inside the radical allows effective simplification and makes arithmetic clearer and more manageable.

Factoring Out Common Terms

Factoring involves identifying common components within an expression and "pulling" them out of the calculation. It simplifies expressions significantly by reducing complexity and eliminating redundant terms.

In our given expression, once simplified, we encountered terms such as \(-25 \sqrt[3]{5}\) and \(2 \sqrt[3]{5}\). Noticing that \(\sqrt[3]{5}\) is present in both allows us to factor it out, combining these terms efficiently. Here's how it works:

• Identify the common term.
• Collect the coefficients (in our exercise, \(-25\) and \(2\)).
• Factor out the common term using the expression: \((-23) \times \sqrt[3]{5}\).

This process reduces potential errors and highlights the underlying simplicity in complex problems, offering a beautiful symmetry to calculations.