Problem 14 - 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. $$ \left(\frac{2 x}{3 y}\right)^{3} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\left(\frac{2x}{3y}\right)^3\). Answer: \(\frac{8x^3}{27y^3}\)

Step 1: Identify the rules to apply

In this exercise, we'll be using the power rule for quotients and the power rule for powers. The power rule for quotients states that \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\). The power rule for powers states that \((a^m)^n = a^{mn}\).

Step 2: Apply the power rule for quotients

Begin by applying the power rule for quotients to the given expression: \(\left(\frac{2x}{3y}\right)^3 = \frac{(2x)^3}{(3y)^3}\).

Step 3: Apply the power rule for powers

Next, we apply the power rule for powers separately to the numerator and denominator: \((2x)^3 = 8x^3\) and \((3y)^3 = 27y^3\).

Step 4: Simplify the expression

Finally, put the simplified numerator and denominator back into a single fraction: $$\frac{8x^3}{27y^3}$$ And this is the simplified expression for the given exercise.

Key Concepts

Power Rule for Quotients

Understanding the power rule for quotients is essential when working with algebraic expressions that involve division. Imagine you have a fraction, where both the numerator and the denominator are taken to the same power. Instead of working with the fraction as a whole, you can apply the power to both parts separately. Mathematically, this rule is expressed as:
\[\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\]
This means if you have something like\(\left(\frac{x}{y}\right)^3\), you can apply the exponent of 3 to both the numerator \(x\) and the denominator \(y\), giving you \(x^3\) and \(y^3\) respectively.

Power Rule for Powers

The power rule for powers helps simplify expressions where you have an exponent raised to another exponent. It's like a stacking effect; you're taking a power of a power. According to this rule:
\[ (a^m)^n = a^{mn} \]
What you do is multiply the two exponents together. For example, if you have \((2^2)^3\), you would multiply the 2 and 3 to get \(2^{2 \times 3} = 2^6\). This exponent multiplication is handy when you are simplifying complex algebraic expressions, especially when variables are also involved. It transforms a seemingly daunting expression into a more manageable one without altering its value.

Algebraic Expressions

Algebraic expressions are the backbone of algebra. They combine constants and variables with arithmetic operations to represent relationships and solve problems. When looking at \(\left(\frac{2x}{3y}\right)^3\), we're dealing with an algebraic expression where \(2x\) and \(3y\) are bound by division, and the entire fraction is raised to a power. Simplifying such expressions often requires us to apply specific exponent rules. Understanding how these rules apply not only helps in simplification but also enhances problem-solving skills for more complex algebraic challenges.

Exponent Rules

There are several exponent rules that are vital when manipulating and simplifying expressions with exponents. These include the power rule for products and quotients, the power rule for powers, the product of powers rule, and the zero exponent rule, among others.
Each of these rules provides a systematic way to simplify expressions that might otherwise seem intricate. For instance, when raising a quotient to a power, we use the power rule for quotients. For an expression where an exponent is raised to another power, we use the power rule for powers. Familiarity with these rules not only makes algebra more approachable but also more fascinating as it unveils patterns and simplifies complex expressions into more understandable forms.