Problem 83 - 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: Factor Out Perfect Cubes for \( \sqrt[3]{125} \)

First, we need to break \( 125 \) into prime factors to determine its cube root. We know that \( 125 = 5 \times 5 \times 5 = 5^3 \). So, the cube root \( \sqrt[3]{125} \) simplifies to 5.

Step 2: Simplify \( 2 \sqrt[3]{125} \)

With the result from Step 1, substitute \( \sqrt[3]{125} = 5 \) back into the expression which gives \( 2 \times 5 = 10 \).

Step 3: Factor Out Perfect Cubes for \( \sqrt[3]{64} \)

Next, we break \( 64 \) into prime factors to find its cube root. Notice that \( 64 = 4 \times 4 \times 4 = 4^3 \), so \( \sqrt[3]{64} = 4 \).

Step 4: Simplify \( 5 \sqrt[3]{64} \)

Using the result from Step 3, substitute \( \sqrt[3]{64} = 4 \) back into the expression which gives \( 5 \times 4 = 20 \).

Step 5: Combine the Simplified Terms

Subtract the result from Step 4 from the result of Step 2: \( 10 - 20 = -10 \).

Key Concepts

Simplification

Simplification in algebra involves reducing expressions to their simplest form while maintaining their original value. This process often makes calculations easier and solutions clearer. Simplifying expressions can involve a variety of techniques, from combining like terms to reducing fractions or factoring.

• Like Terms: These are terms in an expression that contain the same variables raised to the same power. For example, in the expression \(2x + 3x\), both terms can be added to simplify it to \(5x\).
• Factoring: Sometimes, simplifying involves factoring out the greatest common factors. This means dividing different parts of the expression by their common factors to simplify.
• Reducing Fractions: Another common simplification technique is reducing fractions by dividing both the numerator and the denominator by their greatest common divisor.

By carefully applying these methods, we can transform complicated or lengthy equations into more manageable forms, making calculations and further algebraic manipulations straightforward.

Cube Roots

Cube roots are the opposite operation of cubing a number. The cube root of a number \(n\) is a value \(x\) such that \(x^3 = n\). For example, if \(n = 27\), then the cube root is 3, since \(3^3 = 27\).

• Notation: The cube root of \(n\) is written as \(\sqrt[3]{n}\).
• Finding Cube Roots: To find the cube root of a number, identify if the number is a perfect cube. This means it can be expressed as \(a^3\), where \(a\) is an integer.
• Use of Prime Factorization: If a number is not immediately obvious as a perfect cube, it can be useful to perform prime factorization and express it in terms of its prime factors.

Cube roots are crucial in algebra for simplifying expressions and solving equations where variables are raised to the power of three. Recognizing and finding cube roots quickly can streamline problem-solving significantly.

Perfect Cubes

Perfect cubes are numbers that can be expressed as an integer raised to the third power, such as \(1, 8, 27, 64, 125\), etc. The concept of perfect cubes is foundational in determining cube roots and simplifying cube root expressions.

• Identifying Perfect Cubes: A perfect cube is always the result of an integer multiplied by itself twice more. For instance, \(4^3 = 64\).
• Examples: Common perfect cubes include \(1\times1\times1 = 1\), \(2\times2\times2 = 8\), and \(3\times3\times3 = 27\).
• Usage in Simplification: Recognizing perfect cubes allows for easy computation of cube roots and significant simplification of algebraic expressions involving cube roots.

Understanding perfect cubes not only aids in solving algebraic problems but also enhances number sense and mathematical pattern recognition, especially when dealing with power operations.