Problem 17 - Elementary Algebra

Question

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. $$ \left(\frac{8 a^{3} b^{2} c^{6}}{4 a^{2} b}\right)^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression $\left(\frac{8^3 a^{3\times3} b^{2\times3} c^{6\times3}}{4^3 a^{2\times3} b^{3}}\right)$ Answer: $8 a^3 b^3 c^{18}$

Step 1: Apply power rule for powers

Raise each term within the brackets to the power of 3. $$ \left(\frac{8^3 a^{3\times3} b^{2\times3} c^{6\times3}}{4^3 a^{2\times3} b^{3}}\right) $$

Step 2: Simplify each term

Simplify each term in both the numerator and denominator. $$ \left(\frac{512 a^{9} b^{6} c^{18}}{64 a^{6} b^{3}}\right) $$

Step 3: Apply power rule for quotients

Divide the like terms in numerator with those in the denominator. $$ \frac{512 a^{9} b^{6} c^{18}}{64 a^{6} b^{3}} = 8 a^{(9-6)} b^{(6-3)} c^{18} $$

Step 4: Simplify final expression

Write down the simplified expression. $$ 8 a^3 b^3 c^{18} $$

Key Concepts

Understanding Quotients

When simplifying expressions involving quotients, it's all about division. A quotient occurs when you divide one expression by another. In mathematics, particularly algebra, the power rule for quotients helps simplify expressions with exponents.
This rule states that when you divide two terms with the same base, you subtract the exponents. For instance, \( \frac{a^m}{a^n} = a^{m-n} \). This is what we apply during Step 3 of the solution: dividing each term in the numerator by the corresponding term in the denominator.

• Reduce coefficients directly, e.g., \( \frac{512}{64} = 8 \).
• Subtract exponents of like bases, so \( \frac{a^9}{a^6} \) becomes \( a^{3} \).
• This also applies to \( b \) with \( b^6 / b^3 = b^3 \).

Therefore, understanding how to divide and reduce similar terms is crucial.

Working with Products

The power rule for products deals with multiplying terms. In an expression, when different variables or numbers are multiplied, we call the result a product. Applying the power rule for products, when elements are within parentheses being raised to a power, you multiply each exponent by that power.

In Step 1, for instance, each term in \( (8a^3b^2c^6) \) is raised to the power of 3.

• Calculate \( 8^3 \), which results in 512.
• Multiply each exponent by 3: \( a^{3 \times 3} = a^9 \).
• Repeat for \( b \) and \( c \), resulting in \( b^6 \) and \( c^{18} \).

This method allows for simplifying complex expressions involving multiple terms and operations.

Exploring Powers

Powers refer to exponents, which indicate how many times a number, called the base, is multiplied by itself. The power rule for powers, specifically, simplifies when a power is raised to another power. In this case, you multiply the exponents.
In our solution, this is seen in Step 1 when each term in the expression is raised to the power of 3.

• Consider \( a^{(3 \times 3)} \) which is simplified to \( a^9 \).
• This rule supports handling expressions like \( (x^m)^n = x^{m \times n} \).

By multiplying exponents, you efficiently break down expressions with nested powers.

Steps to Simplification

Simplification reduces expressions to their simplest form. It’s akin to cleaning up messy stacks of terms until they are straightforward and easy to interpret.
For our expression, simplification unfolds in these steps:

• First, apply rules for powers and products, multiplying where needed.
• Next, handle division with the power rule for quotients, simplifying like bases.
• Finally, finess steps using arithmetic to reduce numerical coefficients.

After following the rules methodically, the expression transforms from complex to simple: \( 8a^3b^3c^{18} \).
Simplification is key in algebra to provide clear and concise results.