Problem 61 - Elementary Algebra

Question

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$ \left(\frac{3 a b}{4 x y}\right)^{3} $$

Step-by-Step Solution

Answer

Answer: The simplified expression is $\frac{27 a^3 b^3}{64 x^3 y^3}$.

Step 1: Distribute the exponent to each term inside the parentheses

Apply the power rule to distribute the exponent of 3 to each term inside the parentheses. Recall that (a * b)^n = a^n * b^n: $$ \left(\frac{3 a b}{4 x y}\right)^{3} = \frac{(3 a b)^3}{(4 x y)^3} $$

Step 2: Apply the power rule to each term in the numerator and denominator of the fraction

Now, apply the power rule to each term in the numerator and the denominator of the fraction: $$ \frac{(3 a b)^3}{(4 x y)^3} = \frac{3^3 \cdot a^3 \cdot b^3}{4^3 \cdot x^3 \cdot y^3} $$

Step 3: Calculate the powers of the constants and rewrite the expression

Compute the powers of the constants 3 and 4, and rewrite the expression with the simplified fractions: $$ \frac{3^3 \cdot a^3 \cdot b^3}{4^3 \cdot x^3 \cdot y^3} = \frac{27 a^3 b^3}{64 x^3 y^3} $$ The simplified expression is: $$ \left(\frac{3 a b}{4 x y}\right)^{3} = \frac{27 a^3 b^3}{64 x^3 y^3} $$

Key Concepts

Power Rules for Exponents

When working with exponents, understanding the power rules, also known as the laws of exponents, is crucial to simplifying expressions effectively. The power rules are a set of guidelines that describe how to manipulate exponential terms. One of the key rules is the power of a power rule, which states that when you raise an exponential expression to another exponent, you multiply the exponents. Mathematically, it's presented as \( (x^a)^b = x^{a*b} \).

For example, if we have \( (2^2)^3 \), following the power rule, we would multiply the exponents 2 and 3, resulting in \( 2^{2*3} \), which simplifies to \( 2^6 \). This rule is essential when an entire fraction is raised to an exponent, as you'll see in the textbook problem. Each term within the parentheses—both numerator and denominator—is raised to the third power, indicating application of this power rule.

Distributive Property of Exponents

Besides the power rules, the distributive property of exponents is another vital concept. This property allows you to distribute an exponent over terms that are multiplied together within a parenthesis. When you have \( (abc)^n \), it distributes to \( a^n * b^n * c^n \). It's important to remember that this property only works with multiplication and not with addition or subtraction inside the parentheses.

In the given exercise, after distributing the exponent of 3 to each term in \( (3ab)^3 \) and \( (4xy)^3 \), we get \( 3^3 * a^3 * b^3 \) and \( 4^3 * x^3 * y^3 \) respectively. This step-by-step breakdown demonstrates the distributive property in action, transforming the expression into one that is much easier to compute and understand.

Simplifying Algebraic Fractions

Finally, simplifying algebraic fractions is a process of reducing expressions to their simplest form. In the context of the exercise, once the power and distributive properties have been applied, we're left with a fraction where the numerator and denominator have exponents that can be evaluated. Simplifying the coefficients—like \( 3^3 \) and \( 4^3 \)—and any variables with exponents, results in an expression free of exponents.

The final step involves computing the powers of constants. In our exercise, \( 3^3 = 27 \) and \( 4^3 = 64 \) are calculated to simplify the fraction further, resulting in \( \frac{27 a^3 b^3}{64 x^3 y^3} \). This process of breaking down and simplifying each component is what makes algebraic fractions manageable, ultimately leading to the simplest form of the expression.