Problem 127 - Elementary Algebra

Question

For the following problems, write each expression so that only positive exponents appear. $$ \left(y^{-3}\right)^{-4} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(\left(y^{-3}\right)^{-4}\) using only positive exponents. Answer: \(y^{12}\)

Step 1: Recall the rules of exponents

Before we start working with the expression, let's recall two essential rules of exponents: 1. \(a^{m \cdot n} = \left(a^m\right)^n\) 2. \(a^{-m} = \frac{1}{a^m}\) In this problem, we are given an expression with negative exponents, and we can use these rules to simplify the expression.

Step 2: Apply the rules of exponents

Now we will apply the rules of exponents to the given expression: $$ \left(y^{-3}\right)^{-4} $$ Using the first rule: $$ y^{-3 \cdot (-4)} $$ Now, simplify the exponent: $$ y^{12} $$ As the final expression has a positive exponent, we don't need further simplification. The final answer is: $$ y^{12} $$

Key Concepts

Negative Exponents

Understanding the concept of negative exponents is crucial in the study of algebra. A negative exponent represents the reciprocal of the base raised to the absolute value of the exponent. In other words, when you see an expression like \( x^{-n} \), it is equivalent to \( \frac{1}{x^n} \), where \( n \) is a positive integer. This rule helps us transform an expression with a negative exponent into one with a positive exponent, making it easier to work with.

For example, consider the expression \( y^{-3} \). According to the negative exponent rule, this is the same as \( \frac{1}{y^3} \), which reveals the inverse relationship between positive and negative exponents. When simplifying expressions, it's important to remember that a negative sign in the exponent flips the position of the base—in this case, \( y \) moves from the numerator to the denominator of a fraction.

Rules of Exponents

The rules of exponents are mathematical shortcuts that allow us to simplify expressions involving powers of the same base. Some of the fundamental rules include the product rule \(a^m \cdot a^n = a^{m+n}\), the power rule \((a^m)^n = a^{m \cdot n}\), and of course, the rule for negative exponents that we previously discussed.

Using these rules, we can handle complex expressions like \( \left(y^{-3}\right)^{-4} \). Applying the power rule, which states that you can multiply the exponents when a power is raised to another power, we get \( y^{-3 \cdot (-4)} = y^{12} \), effectively transforming a negative exponent into a positive one. It's essential to grasp these rules thoroughly, as they form the basis for solving a wide array of algebraic problems, ensuring that expressions are simplified correctly and efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a specific mathematical relationship. They can range from simple, like \( 2x + 3 \), to complex, involving multiple terms and powers, like \( 3x^2y - y^{-3} + 4z \). The beauty of algebra lies in how these expressions can be manipulated using a set of rules and properties to reveal underlying patterns or solve equations.

An essential skill in working with algebraic expressions is simplification, which often involves applying the rules of exponents. Simplification makes expressions more manageable and prepares them for further operations such as addition, subtraction, or solving equations. Remember, the goal is to express algebraic terms as succinctly as possible without changing the expression's value. Proper simplification can often be the key to unlocking the solution to an algebraic problem.