Simplifying Algebraic Expressions: Techniques and Steps

Algebraic expressions can often appear complex, making it difficult to understand their true values and relationships. Simplifying these expressions can help in solving equations and analyzing mathematical functions more efficiently. One such problem involves the expression left( frac{27a^9}{125b^3} right)^{-frac{2}{3}}. This article will guide you through the process of simplifying this expression step-by-step, using fundamental algebraic rules and properties of exponents.

Step-by-Step Simplification

Let's begin by understanding the given expression: left( frac{27a^9}{125b^3} right)^{-frac{2}{3}}.

Apply the Negative Exponent

Recall that x^{-n} frac{1}{x^n}. This property can be applied to the given expression:

left( frac{27a^9}{125b^3} right)^{-frac{2}{3}} frac{1}{left( frac{27a^9}{125b^3} right)^{frac{2}{3}}}

Simplify the Fraction Inside the Exponent

The next step involves simplifying the fraction inside the exponent. We can rewrite the expression as:

left( frac{27a^9}{125b^3} right)^{frac{2}{3}} frac{27^{frac{2}{3}}a^9^{frac{2}{3}}}{125^{frac{2}{3}}b^3^{frac{2}{3}}}

Calculate Each Component

Now, we calculate each individual component:

27^{frac{2}{3}} 3^3^{frac{2}{3}} 3^2 9 a^9^{frac{2}{3}} a^{9 cdot frac{2}{3}} a^6 125^{frac{2}{3}} 5^3^{frac{2}{3}} 5^2 25 b^3^{frac{2}{3}} b^{3 cdot frac{2}{3}} b^2

Substituting these results back into the expression gives:

left( frac{27a^9}{125b^3} right)^{frac{2}{3}} frac{9a^6}{25b^2}

Combine the Results

Finally, we substitute this result back into the initial expression with the negative exponent:

frac{1}{left( frac{27a^9}{125b^3} right)^{frac{2}{3}}} frac{1}{frac{9a^6}{25b^2}} frac{25b^2}{9a^6}

Therefore, the simplified form of the expression left( frac{27a^9}{125b^3} right)^{-frac{2}{3}} is frac{25b^2}{9a^6}.

Alternative Method

Given the identity left( frac{a}{b} right)^c frac{a^c}{b^c}, we can also simplify the expression by first calculating the numerator and the denominator separately before combining them. This approach can be particularly useful in more complex expressions.

Let's break down the given expression:

27a^9^{-2/3} frac{1}{27a^9^{2/3}} frac{1}{left( sqrt[3]{27a^9} right^2} frac{1}{3a^3^2} frac{1}{9a^6}

125b^3^{-2/3} frac{1}{125b^3^{2/3}} frac{1}{left( sqrt[3]{125b^3} right^2} frac{1}{5b^2} frac{1}{25b^2}

Now, combining these results using the division rule of exponents: frac{a}{b} frac{a}{b} times frac{d}{c} frac{a times d}{b times c}. This gives:

left( frac{27a^9}{125b^3} right)^{-frac{2}{3}} frac{frac{1}{9a^6}}{frac{1}{25b^2}} frac{1}{9a^6} times 25b^2 frac{25b^2}{9a^6} left( frac{5b}{3a^3} right)^2

While manual simplification can be valuable, it's also important to recognize when a computer algebra system can assist in the process, especially with more complex expressions. It is generally assumed that a, b in R^ when simplifying in this manner.

Conclusion

Simplifying algebraic expressions, such as left( frac{27a^9}{125b^3} right)^{-frac{2}{3}}, utilizing exponent rules and properties allows us to break down complex expressions into more manageable forms. This not only enhances mathematical comprehension but also aids in problem-solving and analysis. By understanding these techniques, you can approach similar problems with confidence.