Problem 57 - Algebra A Combined Function

Question

Simplify. Assume that all variables represent positive real numbers. $$ \sqrt[3]{125 r^{9} s^{12}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 5r^3s^4 \).

Step 1: Simplify the Cube Root of the Constants

First, extract the cube root of the coefficient 125. Since 125 is a perfect cube, calculate \( \sqrt[3]{125} = 5 \). This simplifies the problem to \( 5 \sqrt[3]{r^{9} s^{12}} \).

Step 2: Simplify the Cube Root of the Variable \(r^9\)

For \( r^9 \), use the property \( \sqrt[3]{r^9} = r^{9/3} = r^3 \). Therefore, the expression becomes \( 5r^3 \sqrt[3]{s^{12}} \).

Step 3: Simplify the Cube Root of the Variable \(s^{12}\)

Similarly, for \( s^{12} \), use the property \( \sqrt[3]{s^{12}} = s^{12/3} = s^4 \). Now the expression simplifies to \( 5r^3s^4 \).

Step 4: Final Combined Expression

Combine all the simplified parts together: The final simplified expression is \( 5r^3s^4 \).

Key Concepts

Simplifying Cube Roots

Cube roots can seem tricky at first, but with some practice, you will find them quite manageable. A cube root asks the question: "What number can be multiplied by itself three times to get the original number?"

To simplify cube roots, look for "perfect cubes" within the expression. Perfect cubes are numbers like 1, 8, 27, 64, and 125, because their cube roots are whole numbers. When simplifying, separate the expression into perfect cube parts and other factors.

• Identify perfect cubes in numbers and variables.
• Take the cube root of these perfect cubes.
• Simplify any remaining parts of the expression.

This method helps break down complicated expressions into simpler, more manageable parts.

Perfect Cubes

Understanding perfect cubes is crucial for simplifying cube roots. A perfect cube is a result of a number multiplied by itself twice more, like:

• \( 2 imes 2 imes 2 = 8\)
• \( 3 imes 3 imes 3 = 27\)
• \( 5 imes 5 imes 5 = 125\)

These numbers are "perfect" because they can be neatly snapped back into place by taking the cube root.
Variables can also form perfect cubes. For example, in the expression \( r^9 \), since \( 9 = 3 \times 3 \), \( r^9 \) is a perfect cube of \( (r^3)^3 \). Knowing these allows quick simplification when facing cube roots.

Exponent Rules

Exponent rules help simplify expressions with powers efficiently and systematically. When managing cube roots, these rules become vital:

• The cube root of a power, like \( x^{12}\), involves dividing the exponent by three: \( x^{12/3} = x^4 \).
• This works due to the fundamental property of roots and powers, where \((x^m)^n = x^{m\times n }\).

When simplifying cube roots, remember:

• The root operation can be applied to both coefficients and variables separately.
• Check if the powers in variables result in whole numbers after division, helping identify perfect cubes.

These rules streamline the process of breaking down expressions cleanly and effectively.

Real Numbers

Real numbers are the cornerstone of many mathematical concepts, including cube roots. Real numbers include all the rational numbers like integers, fractions, and the irrational numbers.

In calculus and algebra, we assume, like in the original problem, that all variables signify positive real numbers, which means:

• The cube roots and other root operations will yield real, sensible results.
• There is no concern over complex numbers in these contexts, making calculations straightforward.

The understanding of real numbers varies with real-life examples and theoretical concepts, all governed by real number properties, underscoring their integral role in math problems.