Problem 46 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \left(\frac{8}{x}\right)^{-2} $$

Step-by-Step Solution

Answer

The simplified form of the original expression \[ \left(\frac{8}{x}\right)^{-2} \]is \[ \frac{x^2}{64}\]

Step 1: Understand how negative exponents work

The first step is to understand how negative exponents work. A fraction raised to a negative exponent, the base of the exponent can be flipped or inverted. Hence, \[ \left(\frac{8}{x}\right)^{-2}\]becomes \[ \left(\frac{x}{8}\right)^{2}\]

Step 2: Apply the power to the numerator and the denominator

Now, apply the power individually to the numerator and the denominator. Which gives us \[ \left(\frac{x}{8}\right)^{2} = \frac{x^2}{8^2}\]

Step 3: Simplify the denominator

Now that we've applied the power, all we have left to do is to simplify the denominator (8^2). This simplifies to 64. Thus, our final result is \[ \frac{x^2}{64}\]

Key Concepts

Simplifying Expressions

Simplifying expressions might sound complex, but it's all about making the expression as straightforward as possible. It's like organizing a messy room! You aim to reduce clutter so everything is neat and tidy.

When simplfying algebraic expressions, we often deal with things like variables, coefficients, and exponents. The goal is to combine like terms and eliminate any unnecessary complexity. In our exercise, simplifying involves understanding negative exponents and applying them to the expression.

By flipping the fraction when encountering a negative exponent, you essentially rewrite the expression to a positive power, making it simpler and easier to handle. Once you've rewritten the expression with positive exponents, it's much easier to multiply or further simplify it. This approach helps tidy up the expression so you can clearly see and understand each part of it.

Powers and Exponents

Powers are a shorthand way of showing how many times a number is multiplied by itself. For example, if you have \(3^2\), it means \(3\times3\). Exponents are tiny numbers written above and to the right of a base number to indicate its power. They are powerful tools in math and are used in many calculations.

When handling powers and exponents, it's important to remember the rules of exponents. A negative exponent means you take the reciprocal, or flip the base, and then apply the positive power. For instance, \(a^{-n} = \frac{1}{a^n}\).

In solving the exercise, we encountered \(\left(\frac{8}{x}\right)^{-2}\). According to exponent rules, we flip the fraction, resulting in a positive power. This makes calculations easier and keeps the numbers straightforward. By understanding these rules, you can effectively manage and simplify expressions with exponents.

Algebraic Fractions

Algebraic fractions involve expressions with variables in the numerator, denominator, or both. Think of them as regular fractions but with a twist of algebra!

Simplifying algebraic fractions often entails dealing with the numerator and denominator separately to make the expression as simple as possible. The exercise demonstrates this as you apply powers to \(x\) and 8 individually.

Here, the initial fraction is raised to a power, so we apply the power to both the numerator and the denominator. You calculate \(x^2\) and \(8^2\), which gives us \(\frac{x^2}{64}\). This method ensures each part of the fraction is simplified correctly.

Understanding algebraic fractions, along with the rules of simplifying them, is crucial for solving exercises like this one. By practicing, you become more comfortable with manipulating expressions and finding the simplest form.