Problem 28 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (x-5)^{-3} $$

Step-by-Step Solution

Answer

Answer: \(\frac{1}{(x-5)^3}\)

Step 1: Rewrite the expression with a positive exponent

Recall that a negative exponent can be rewritten as the reciprocal of the base raised to the positive value of the exponent. In our case, the base is \((x-5)\) and the exponent is \(-3\). The reciprocal of \((x-5)\) is \(\frac{1}{(x-5)}\). Therefore, we can rewrite the expression as: $$ \left(\frac{1}{(x-5)}\right)^3 $$

Step 2: Simplify the expression

Now, we will simplify the expression by raising the term inside the parentheses to the power of 3. For this, we need to raise both the numerator and denominator separately to the power of 3: $$ \left(\frac{1}{(x-5)}\right)^3 = \frac{1^3}{(x-5)^3} $$ Since \(1^3 = 1\), the expression becomes: $$ \frac{1}{(x-5)^3} $$ That's our final expression using only positive exponents.

Key Concepts

Understanding Negative Exponents

Negative exponents often confuse students, but the concept is quite simple when broken down. When you encounter a negative exponent in an expression, it indicates that you need to take the reciprocal of the base.
For example, if we have \(a^{-n}\), it becomes \(\frac{1}{a^n}\).

• The base \(a\) switches from the numerator to the denominator.
• The exponent changes from negative to positive.

This shift does not change the overall value of the expression; it merely expresses it in another form. In our exercise, the base \((x-5)^{-3}\) becomes \(\left(\frac{1}{(x-5)}\right)^3\), transforming the negative exponent into a manageable positive one.

The Reciprocal Explained

Understanding the reciprocal is essential for simplifying expressions with negative exponents.
The reciprocal of a number is simply one divided by that number.

• For example, the reciprocal of \(a\) is \(\frac{1}{a}\).
• If the base was a single number or variable, it would be directly flipped.

In our case, the reciprocal of \((x-5)\) is \(\frac{1}{(x-5)}\).
This transformation is the first key step to simplifying expressions with negative exponents into those with positive ones. Understanding this process will help unravel many other algebraic challenges.

What is an Expression in Algebra?

An algebraic expression is a combination of numbers, variables, and operators like addition and subtraction. In our situation, we focused on the expression \((x-5)^{-3}\).
Expressions can be simple or complex, involving multiple steps and operations.

• Defined by mathematical operations between elements.
• They can be simplified by following standard algebraic rules.

The ability to manipulate expressions—such as transforming negative exponents into positive ones—brings clarity to what might initially seem complicated.

Simplifying Expressions

Simplification is the process of rewriting an expression in its simplest form. It's about reducing complexity without changing the expression's value.
For instance,

• In \((x-5)^{-3}\), the goal was to express it with positive exponents.
• We achieved this by switching the base's position in the fraction.

After addressing the exponent, we were left with \(\frac{1}{(x-5)^3}\), a neatly simplified expression.
Simplification helps not just in solving algebraic problems but also in making expressions easier to understand and work with across different mathematical contexts.