Question

Rewrite the expression with positive exponents and simplify.\(\left(\frac{x^{-3} y^{4}}{5}\right)^{-3}\)

Short answer

The simplified form of the given expression with positive exponents is \( \frac{x^{9} y^{12}}{125} \)

Step-by-step solution

Apply Power of Negative Power Rule

In this step, apply the power-of-negative-power rule which states that \( (a^{-m})^n \) is equal to \( a^{-mn} \). Therefore, reverse the signs of the exponents in the expression. This gives us \( \left(\frac{x^{3*-(-3)} y^{4*-(-3)}}{5^{-(-3)}}\right) \).

Simplify the Exponents

Simplify the exponents in the expression obtained which results in \( \left(\frac{x^{9} y^{12}}{125}\right) \).

Final Simplification

The final simplified expression with positive exponents is \( \frac{x^{9} y^{12}}{125} \)

Key concepts

Power of Negative Power Rule

The Power of Negative Power Rule is a fundamental principle in algebra used to simplify expressions involving exponents. According to this rule, when you take an exponent to another exponent, and the base exponent is negative, you multiply the two exponents. Significantly, the rule states that \( (a^{-m})^n = a^{-mn} \), which essentially flips the sign when you apply the power if both the initial exponent and the one you're raising it to are negative.

This rule is very helpful when dealing with complex expressions where negative exponents are raised to other powers. The aim is to end up with positive exponents because they are much easier to handle. For example, if you have the expression \( (x^{-2})^3 \), applying the power of a negative power rule would result in \( x^{-6} \), which is equivalent to \( \frac{1}{x^6} \). This simplification process is essential for students as it paves the way towards understanding more advanced concepts in algebra and calculus.

Exponent Simplification

Simplification of exponents is an act of reducing expressions to their simplest form while adhering to the laws of exponents. When simplifying exponents, it's crucial to keep in mind several rules, such as the power rule, product rule, and quotient rule, among others. One of the key processes during simplification is to convert any negative exponents into positive ones because it makes the expression easier to comprehend and manipulate.

In simplifying exponents, you should always aim to perform all possible operations: combine like terms, apply the power of a negative power rule when necessary, and reduce any fractional exponents to their simplest form. For instance, in the expression \( x^{9} \), there are no further simplification steps needed. However, if we have \( x^{2} \cdot x^{3} \), we can simplify it to \( x^{5} \), using the rule that allows us to add exponents when multiplying like bases. This step-by-step approach aids learners in understanding how to navigate through more complex algebraic operations and helps ensure accuracy in their work.

Algebraic Expressions

Algebraic expressions are mathematical phrases that encompass constants, variables, and operational symbols. They can be as simple as a single variable \( x \) or as intricate as polynomials like \( 4x^2 + 3xy - 5 \). When dealing with algebraic expressions, one of the goals is to manipulate them into a simpler or more useful form using various algebraic rules.

Key steps when working with algebraic expressions include combining like terms, factoring, expanding, and simplifying exponents as demonstrated in previous sections. Understanding how to work with exponents is particularly essential, as they frequently appear in algebraic expressions. Moreover, mastering the manipulation of algebraic expressions is crucial for solving equations and inequalities, which are foundational elements of algebra.

By strengthening their skills in simplifying algebraic expressions, students enhance their ability to tackle a wide range of mathematical problems. They learn to see patterns, make predictions, and solve real-world problems, translating abstract concepts into actionable solutions.