Problem 83 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ 2 \sqrt[3]{125}-5 \sqrt[3]{64} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-10\).

Step 1: Simplify the Radical Terms

First, we simplify each cube root in the expression. Let's start with \(\sqrt[3]{125}\). Since 125 is \(5^3\), we have \(\sqrt[3]{125} = 5\). Next, let's simplify \(\sqrt[3]{64}\). Since \(64 = 4^3\), we have \(\sqrt[3]{64} = 4\).

Step 2: Substitute Simplified Values

Substitute the simplified values back into the original expression. This gives us: \[2 \cdot 5 - 5 \cdot 4\]

Step 3: Perform Multiplication

Calculate the multiplication for each term. \[2 \times 5 = 10 \] and \[5 \times 4 = 20\].

Step 4: Subtract the Results

Finally, subtract the second result from the first. \[10 - 20 = -10\].

Key Concepts

Cube Roots

Understanding cube roots is crucial when working with expressions involving radical terms. A cube root asks us to find a number that, when multiplied by itself three times, equals the original number under the root.

• For example, the cube root of 125 (\( \sqrt[3]{125} \)) is 5 because 5 multiplied by itself three times (\(5 \times 5 \times 5\)) gives us 125.
• Similarly, to find the cube root of 64 (\( \sqrt[3]{64} \)), we look for a number that gives 64 when cubed. This number is 4, as \(4 \times 4 \times 4 = 64\).

By identifying the base numbers in cube roots, you will simplify the radicals into whole numbers, which makes algebraic manipulation easier.

Radical Expressions

Radical expressions involve roots, such as square roots and cube roots, and are encountered frequently in algebra. When simplifying radical expressions, the goal is to reduce them to their simplest form by resolving the roots.

• A radical expression like \(2 \sqrt[3]{125}\) requires breaking down the cube root based on known powers. Once we simplify \(\sqrt[3]{125}\) to 5, the expression becomes easier to work with as \(2 \times 5\).
• The key is identifying perfect cubes or simplifying the radicals to reduce expressions for easier calculation.

Working through radical expressions step-by-step, as shown in the solution, ensures each component is simple for subtraction or addition, if necessary.

Algebraic Simplification

Algebraic simplification involves reducing expressions into their most basic form without changing their value or meaning. This requires understanding and applying mathematical operations such as multiplication, division, and subtraction.

• First, we reduce complex expressions by simplifying each component. For example, in the expression \(2 \sqrt[3]{125} - 5 \sqrt[3]{64}\), the cube roots were simplified to integers.
• Then we substitute these simplified values to make the expression easier to manage, resulting in \(2 \times 5\) and \(5 \times 4\).

The final step in simplification is performing these mathematical operations and arriving at a single numerical answer. The entire process ensures that complex expressions can be calculated swiftly without losing accuracy, leading to improved problem-solving skills in algebra.