Problem 4 - Elementary Algebra

Question

Write each of the following using only positive exponents. $$ 5^{-1} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression using only positive exponents: 5^-1 Answer: The expression with only positive exponents is: 5^-1 = 1/5.

Step 1: Identify the negative exponent

We are given the expression:$$5^{-1} $$There's a negative exponent of -1, so we will use the rule for negative exponents to rewrite this expression using a positive exponent.

Step 2: Apply the rule for negative exponents

Applying the rule for negative exponents \(a^{-n} = \frac{1}{a^n}\), where a is the base and -n is the negative exponent, we get:$$5^{-1} = \frac{1}{5^1}$$

Step 3: Simplify the result

Now we can simplify the expression:$$\frac{1}{5^1} = \frac{1}{5}$$ The expression with only positive exponents is:$$5^{-1} = \frac{1}{5}$$

Key Concepts

Positive Exponents

Positive exponents are a fundamental concept in mathematics, representing how many times a base number is multiplied by itself. Unlike negative exponents, which indicate division, positive exponents translate into simple multiplication of the base number. For example, in the expression \(5^3\), the number 5 is multiplied by itself three times: \(5 \times 5 \times 5\).

• A positive exponent tells you the number of times to use the base as a factor in a multiplication.
• Expressions like \(a^n\) mean the base \(a\) is multiplied by itself \(n\) times.
• Conversion from negative to positive exponents can help simplify expressions and calculations.

In the original exercise, converting \(5^{-1}\) to positive exponents, we acknowledge that raising the base to the power of 1 is a straightforward display of the concept. Hence, \(5^{1}\) is simply 5.

Exponent Rules

Understanding exponent rules is crucial for handling expressions with both positive and negative exponents. One essential rule is that any number raised to the power of zero equals one, such as \(a^0 = 1\) when \(a eq 0\).The rule used in the exercise is the negative exponent rule, which states that \(a^{-n} = \frac{1}{a^n}\). This indicates that a negative exponent signifies a reciprocal. For example, \(5^{-1}\) becomes \(\frac{1}{5^1}\).Other important rules include:

• Product of powers rule: \(a^m \times a^n = a^{m+n}\)
• Power of a power rule: \((a^m)^n = a^{m \cdot n}\)
• Power of a product rule: \((ab)^n = a^n \cdot b^n\)

These rules provide a toolkit for simplifying and solving algebraic expressions involving exponents.

Algebra Expressions

Algebra expressions form the backbone of algebra and involve combinations of variables, numbers, and operators. These expressions can include exponents, which serve as a shorthand notation for multiplication.When dealing with algebraic expressions, especially those with exponents, it's important to:

• Recognize the role of variables and constants.
• Apply exponent rules to simplify or manipulate the expressions.
• Understand the effect of negative and positive exponents on these expressions.

For example, transforming \(5^{-1}\) in the exercise is a simple operation in algebra but vital for understanding how exponents function within larger expressions. We rewrite the expression with a positive exponent to better integrate it into broader calculations or simplify the problem-solving process.Algebra expressions enable us to generalize patterns and formulate solutions to complex problems across mathematics and applied fields.