Problem 96 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[3]{\frac{b^{3} c^{8}}{125 c^{5}}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{bc}{5} \).

Step 1: Simplify the Fraction Inside the Radical

Inside the cube root, we have the fraction \( \frac{b^{3} c^{8}}{125 c^{5}} \). Simplify this fraction by cancelling out the common \( c^5 \) term. Dividing the numerator and the denominator by \( c^5 \), the expression simplifies to \( \frac{b^3 c^3}{125} \).

Step 2: Apply the Cube Root to Each Part

The expression within the cube root is now \( \frac{b^3 c^3}{125} \). Apply the cube root to both the numerator and the denominator separately: \( \sqrt[3]{b^3 c^3} \) and \( \sqrt[3]{125} \).

Step 3: Simplify the Cube Roots

\( \sqrt[3]{b^3 c^3} = bc \) because the cube root of \( b^3 \) is \( b \) and the cube root of \( c^3 \) is \( c \). \( \sqrt[3]{125} = 5 \) because \( 5^3 = 125 \).

Step 4: Form the Final Answer

Combine the results from Step 3 to form the simplified expression: \( \frac{bc}{5} \).

Key Concepts

Cube Root

The cube root is essentially looking for a number that, when multiplied by itself three times, results in the original number. It is represented by the symbol \( \sqrt[3]{} \). Unlike square roots that "deal" with pairs, cube roots "work" with triples.

In the expression given \( \sqrt[3]{\frac{b^{3} c^{8}}{125 c^{5}}} \), we focus on examining each part under the cube root individually:

• The cube root of a variable raised to the third power, such as \( b^3 \), is simply the variable itself, here \( b \).
• Similarly, \( c^3 \) under a cube root becomes \( c \) because the cube root and the exponent "cancel" each other out.
• For the number 125, we identify a value that when cubed gives 125. Since \( 5^3 = 125 \), \( \sqrt[3]{125} = 5 \).

By understanding cube roots in this context, you can easily simplify expressions involving them.

Fraction Simplification

Fraction simplification helps us make expressions easier to work with. The goal is to reduce fractions to their simplest form by eliminating common factors in the numerator and the denominator.

In our original fraction \( \frac{b^{3} c^{8}}{125 c^{5}} \), notice the presence of \( c^5 \) both in the numerator and denominator. You can divide both parts by \( c^5 \) to simplify it to \( \frac{b^3 c^3}{125} \).

• This simplification step leads to understanding what remains under the cube root after removing unnecessary parts.
• The key here is recognizing and cancelling common terms to focus only on essentials, helping streamline the process.

This not only builds your problem-solving efficiency but also strengthens algebra fundamentals.

Positive Real Numbers

Understanding positive real numbers is crucial since all variables in this context are positive. Positive real numbers are greater than zero and can be fractions, whole numbers, or irrational numbers, as long as they don't include negative figures or imaginary numbers like \( i \) (the square root of -1).

Working under the assumption that variables are positive allows simplifications, like taking roots, to follow smoothly without extra considerations about sign changes.

• Positive numbers remain positive through operations such as squaring or cubing, thus simplifying calculations.
• Similarly, even when roots are taken, the outcome is straightforward without needing adjustments or having to handle potentially negative results.

This assumption streamlines many operations in algebra and eases the manipulation of complex expressions in mathematical problems.