Problem 68 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ (x+5)^{2}(x+5)^{-6} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \((x+5)^{2}(x+5)^{-6}\) using only positive exponents. Answer: \(\frac{1}{(x+5)^4}\)

Step 1: Identifying the base and exponents

First, identify the common base, which is \((x+5)\), and the exponents in the expression: $$ (x+5)^2 \cdot (x+5)^{-6} $$

Step 2: Apply the Product Rule

Now, add the exponents with the same base: $$ (x+5)^{2 + (-6)} $$

Step 3: Simplify the expression

Simplify the exponents by performing the addition operation: $$ (x+5)^{-4} $$

Step 4: Extra information

Though the expression \((x+5)^{-4}\) has already a negative exponent, it follows the instruction to not have any variable (such as x) with negative exponents. If you'd like to write this expression without any negative exponents at all, you can rewrite it as the reciprocal, which would be: $$ \frac{1}{(x+5)^4}$$

Key Concepts

Exponent Rules

Understanding exponent rules is essential for simplifying expressions involving powers. Exponentiation is a mathematical operation that involves two numbers, the base and the exponent. The exponent, written as a superscript, tells us how many times the base is multiplied by itself. For instance, in the expression \((x+5)^2\), the base is \((x+5)\) and the exponent is 2, indicating that \((x+5)\) is used as a factor twice: \((x+5)\times(x+5)\).

Several key rules govern the manipulation of exponents:

• Product Rule: When multiplying two expressions with the same base, like \((x+5)^2\) and \((x+5)^{-6}\), add the exponents: \( (x+5)^{2 + (-6)} \).
• Quotient Rule: When dividing expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator.
• Power of a Power Rule: When raising an expression to a power, multiply the exponents (e.g., \((x^2)^3 = x^{2\underline{\phantom{xxx}}\cdot3}\)).
• Negative Exponent Rule: An expression with a negative exponent represents the reciprocal of the base raised to the corresponding positive exponent (e.g., \(x^{-3} = \frac{1}{x^3}\)).
• Zero Exponent Rule: Any base (except zero) raised to the power of zero is equal to one (e.g., \(x^0 = 1\)).

Applying these rules correctly will yield an expression in its simplest form, which in the case of our exercise turns the negative exponent into a reciprocal to make all exponents positive.

Simplifying Expressions

The goal of simplifying expressions is to make them as straightforward and digestible as possible, often to facilitate other operations or to clearly convey the underlying mathematical relationship. When dealing with algebraic expressions, simplification generally entails combining like terms, applying exponent rules, and carrying out any arithmetic operations required, all while following the conventional order of operations: parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right).

In our exercise, simplification involves converting a negative exponent to a positive one to avoid working with negative exponents. After applying the product rule and adding the exponents, we obtain \((x+5)^{-4}\). This is then converted to \(\frac{1}{(x+5)^4}\) by the negative exponent rule for it to adhere to the initial instruction of using only positive exponents. This process not only conforms to the rules of algebra but also results in a final expression that's generally easier to interpret and use in subsequent mathematical operations or applied problems.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables (symbols that represent unknown values), and operation signs such as addition, subtraction, multiplication, and division. In contrast to algebraic equations, expressions do not contain an equality sign. They can be simplified, factored, expanded, and manipulated to express relationships in different forms, aiding in solving for unknown values or analyzing mathematical models.

For example, in the exercise \((x+5)^2\) multiplied by \((x+5)^{-6}\), we're dealing with an algebraic expression. It is crucial to handle algebraic expressions with care, especially when multiple terms and complex operations are involved. Simplification makes these expressions more manageable, especially when they're a part of larger equations or formulas. In the context of our exercise, correctly applying the rules discussed earlier has allowed us to change the expression with a negative exponent into a more familiar form with positive exponents, ultimately increasing clarity and making the expression easier to work with in various applications.