Problem 10 - Algebra 1

Question

Rewrite as an expression with positive exponents. $$\frac{1}{(2 x)^{-3}}$$

Step-by-Step Solution

Answer

The expression \(\frac{1}{(2x)^{-3}}\) with positive exponents is \((2x)^3\).

Step 1: Understanding the Problem

The expression is \(\frac{1}{(2x)^{-3}}\). Here, we can see there's a negative exponent, -3 within the brackets.

Step 2: Applying the Rule of Exponents

A number raised to a negative exponent is equivalent to the reciprocal of the number raised to that exponent in positive. Therefore, we apply this rule to the expression. (2x)^{-3} becomes \(\frac{1}{(2x)^3}\). This leaves us with \(\frac{1}{\frac{1}{(2x)^3}}\) as the new expression.

Step 3: Simplifying the Expression

A fraction divided by another fraction is similar to multiplying the first fraction by the reciprocal of the second. Hence, \(\frac{1}{\frac{1}{(2x)^3}}\) simplifies to (1) * \((2x)^3\), which equals to \((2x)^3\).

Key Concepts

Negative Exponents Made Easy

Negative exponents can seem tricky at first, but they're just another way to express fractions. When you have a negative exponent, it means you are dealing with the reciprocal of the base raised to the corresponding positive exponent. Let's break it down:

• Negative exponents 'flip' the base to make it a fraction.
• For example, \((2x)^{-3}\) becomes \(\frac{1}{(2x)^3}\).

So, whenever you see a negative exponent, just think about flipping the base into the denominator and making the exponent positive. This helps in simplifying expressions and solving algebra problems with clarity.

Understanding Reciprocals

The reciprocal is a core part of working with negative exponents. A reciprocal is simply what you get when you flip a number or expression upside down. If you start with a fraction, to find the reciprocal, you exchange the numerator with the denominator. Here's how it connects to exponents:

• The reciprocal of \((2x)^{-3}\) is \((2x)^3\) because of the negative exponent rule.
• In expressions like \(\frac{1}{(a)^n}\), you multiply by \(a^n\) to simplify it further.

Reciprocals help you transform complex expressions into simpler forms, which is useful in algebraic simplifications and solving equations.

Simplifying Algebraic Expressions

Algebraic expressions can involve variables, numbers, and different operations like addition, multiplication, or even exponents. Simplification means making the expression nicer to look at or easier to use. Here's how you can deal with them:

• Identify parts of the expression that can be combined or simplified, such as fractions or terms with exponents.
• Use properties of exponents to rewrite parts with negative exponents as positive ones.
• Simplify step by step: focus on clearing out the negative exponents first, then handle multiplication or division.

By mastering how to manipulate and simplify these expressions, you can easily solve equations and understand algebra better.