Problem 52 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 7(w+2)^{-2}(w+1)^{3} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression using only positive exponents: $7(w+2)^{-2}(w+1)^3$ Answer: $7(w+2)^2(w+1)^3$

Step 1: Identify the term with a negative exponent

First, we should find the term with a negative exponent in the given expression. In this expression, the term with a negative exponent is \((w+2)^{-2}\).

Step 2: Take the reciprocal of the term with a negative exponent

Now, we will take the reciprocal of this term to get rid of the negative exponent. The reciprocal of \((w+2)^{-2}\) is \((w+2)^{2}\), which has a positive exponent.

Step 3: Replace the term with its reciprocal

We will now replace \((w+2)^{-2}\) in the original expression with its reciprocal \((w+2)^{2}\). The new expression is: $$ 7(w+2)^2(w+1)^3 $$ So, the expression written using only positive exponents is $$ 7(w+2)^2(w+1)^3 $$

Key Concepts

Positive Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same factor. In the case of positive exponents, the exponent indicates how many times you multiply the base number by itself. For example, in the expression \( (w+2)^2 \), the positive exponent 2 tells you to multiply \( w+2 \) by itself, resulting in \( (w+2) \times (w+2) \).

• Positive exponents make calculations straightforward, as they involve standard multiplication.
• No rearrangement of terms is necessary when dealing with positive exponents, unlike negative exponents that require manipulation.

In the exercise, converting from a negative exponent to a positive one simplifies the expression and makes further algebraic operations more accessible.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations (like addition or multiplication). They serve as a way to represent mathematical relationships. For instance, the expression \( 7(w+2)^2(w+1)^3 \) combines constants like 7 and variables like \( w \).

• These expressions can be simplified or expanded depending on the operation and context.
• Operations within algebraic expressions follow the order of operations, which ensures that calculations are performed correctly.

Understanding algebraic expressions is crucial for solving equations, working with inequalities, and analyzing real-world problems. Simplifying expressions with positive exponents strengthens your algebraic skills and aids in tackling more complex problems down the line.

Reciprocal

The reciprocal of a number is simply one over that number. In the context of exponents, the reciprocal is used to transform negative exponents into positive exponents. If you encounter a term like \( (w+2)^{-2} \), its reciprocal is \( (w+2)^2 \) because you are flipping the term so the previously negative exponent becomes positive.

• Reciprocals are helpful in performing division and simplifying terms with negative exponents.
• Understanding reciprocals allows you to translate complex expressions into more manageable forms.

In the given exercise, recognizing and applying the reciprocal effectively transformed a negative exponent into a positive one, simplifying the expression and adhering to the rule of using only positive exponents.