Problem 53 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 2(a-8)^{-3}(a-2)^{5} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression using only positive exponents: $$2(a-2)^5(a-8)^{-3}$$ Answer: $$\frac{2(a-2)^5}{(a-8)^3}$$

Step 1: Identify expressions with negative exponents.

In the given expression, only \((a-8)^{-3}\) has a negative exponent.

Step 2: Apply exponent properties to rewrite the expression with positive exponents.

Using the exponent property \((a)^{-n}=\frac{1}{a^n}\), we can rewrite the given expression as:$$ \frac{2(a-2)^5}{(a-8)^3} $$ The expression is now written using only positive exponents.

Key Concepts

Exponent Properties

When dealing with algebraic expressions, it's essential to understand the properties of exponents. These properties govern how we manipulate and simplify expressions that involve exponential terms. The basic exponent properties include:

• Product of Powers Property: When multiplying two expressions with the same base, add their exponents, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers Property: When dividing two expressions with the same base, subtract the exponents, \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Property: When raising a power to another power, multiply the exponents, \((a^m)^n = a^{m \cdot n}\).
• Power of a Product Property: When raising a product to a power, distribute the exponent to each factor, \((ab)^n = a^n \cdot b^n\).
• Negative Exponent Property: A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent, \(a^{-n} = \frac{1}{a^n}\).

These properties make it possible to rewrite expressions in more manageable forms, which is especially useful in simplifying complex algebraic expressions.

Negative Exponents

Negative exponents may initially seem confusing, but they are quite straightforward if you remember a simple rule. A negative exponent means you take the reciprocal, or the flipped version, of the base raised to the positive of that exponent. This is a powerful tool in mathematics that allows us to express terms in simpler forms.
For example, \(a^{-3} = \frac{1}{a^3}\).This conversion helps transform terms with negative exponents into equivalent terms with positive exponents.
In the original exercise, the term \((a-8)^{-3}\) is simplified by applying this property. It is rewritten as \(\frac{1}{(a-8)^3}\), effectively removing the negative exponent by placing the term in the denominator. Understanding and applying the negative exponent property is crucial in rewriting algebraic expressions using only positive exponents.

Positive Exponents

Positive exponents represent straightforward multiplication of a number or variable by itself. When always positive, expressions are much easier to handle and apply in equations.
Every time you encounter an exponent, it tells you how many times to multiply the base. For instance, \(a^5\) simply means \(a \times a \times a \times a \times a\).
In our exercise context, the expression \((a-2)^5\) already has a positive exponent. Therefore, there is no need to change it. The focus is instead on converting any negative exponents to their positive form using exponent properties, making the expression more accessible.

Simplifying Expressions

Simplifying algebraic expressions is all about making them as straightforward and easy to interpret as possible. It involves reducing complex expressions into simplest terms using arithmetic and algebraic techniques.
In the exercise, simplifying was necessary to turn terms with negative exponents into those with positive exponents by using reciprocal properties.
Once the conversion is done, the expression\(\frac{2(a-2)^5}{(a-8)^3}\)is achieved. This expression, composed only of positive exponents, is much more usable in further calculations or applications.

• Reduce fractions if possible.
• Combine like terms, if any.
• Ensure all exponents are positive.

These steps make the process of completing algebraic operations more efficient and accurate. Remember, a simplified expression is often easier to work with, both for calculations and to understand the underlying math concepts.