Problem 52 - Algebra 2 and Trigonometry

Question

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{(-x)^{-5}}{x^{-3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( -\frac{1}{x^2} \).

Step 1: Identify the Given Expression

We are given the expression \( \frac{(-x)^{-5}}{x^{-3}} \). Our task is to rewrite this with only positive exponents.

Step 2: Apply the Property of Exponents

Use the property of exponents that states \( a^{-m} = \frac{1}{a^m} \). This means \( (-x)^{-5} \) can be rewritten as \( \frac{1}{(-x)^5} \) and \( x^{-3} \) as \( \frac{1}{x^3} \).

Step 3: Simplify the Expression

Transform the expression by converting negative exponents: \( \frac{(-x)^{-5}}{x^{-3}} \) becomes \( \frac{1}{(-x)^5} \times x^{3} \).

Step 4: Simplify Further Using Inverse

Combine terms: \( x^{3} \times \frac{1}{(-x)^5} \) becomes \( \frac{x^3}{(-x)^5} \).

Step 5: Apply Exponent Rules

Use the rule \( \frac{a^m}{a^n} = a^{m-n} \) to simplify: \( \frac{x^3}{(-x)^5} = (-1)^{-5} \times x^{3-5} \).

Step 6: Simplify Final Expression

\( (-1)^{-5} = -1 \) because it's an inverse power, and \( x^{3-5} = x^{-2} \). Thus, the expression simplifies to \( -x^{-2} \).

Step 7: Convert to Positive Exponents

Rewrite \( -x^{-2} \) as \( -\frac{1}{x^2} \) since \( x^{-2} = \frac{1}{x^2} \).

Key Concepts

Understanding Negative Exponents

Negative exponents can seem tricky at first, but they are all about the position of a number within a fraction. A negative exponent indicates that the number should be moved from the numerator to the denominator or vice versa. The magnitude of the exponent tells you how many times to multiply the base number. For instance, in the expression \((-x)^{-5}\), the negative exponent "-5" means you should take \(-x\) to the 5th power and place it in the denominator: \[ \frac{1}{(-x)^5} \]. Likewise, \(x^{-3}\) becomes \(\frac{1}{x^3}\).

• The rule is simple: \(a^{-n} = \frac{1}{a^n}\).
• Apply this to make expressions easier to work with and to keep equations balanced.

Understanding how to interchange negative and positive exponents is essential for simplifying and rearranging algebraic expressions.

Simplifying Expressions with Exponents

Simplifying expressions that contain exponents involves rewriting them in a more manageable form without losing their value. It involves several steps, like converting negative exponents and combining terms using multiplication and division of powers. When simplifying, ensure your answer uses only positive exponents because they are easier to interpret and work with. In our exercise, we took the expression \(\frac{(-x)^{-5}}{x^{-3}}\) and transformed it step-by-step:

• First, replace negative exponents with fractions: \(\frac{1}{(-x)^5} \times x^3\).
• Then move to a single fraction operation: \(\frac{x^3}{(-x)^5}\).

It's like tidying up a messy room: each step clears clutter and creates a neater, more organized space.

Properties of Exponents

The properties of exponents are like rules to follow that make calculations easier. These include multiplying powers, dividing powers, and handling powers to another power. By understanding these properties, you can break down complex calculations into simpler parts. Let's explore some key properties relevant to our example:

• Power of a Product: \((ab)^m = a^m \times b^m\).
• Product of Powers: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\). In our solution, this was vital for simplifying \(\frac{x^3}{(-x)^5}\) to \((-1)^{-5} \times x^{-2}\).

Applying these properties is like having a toolbox full of perfect tools just when you need them. They simplify complex expressions and help us reach the most reduced form of an algebraic expression efficiently.