Problem 9 - Elementary Algebra

Question

Write each of the following so that only positive exponents appear. $$ x^{-4} y^{7} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $$x^{-4}y^7$$ with only positive exponents. Answer: The expression with only positive exponents is $$\frac{1}{x^4} y^7$$.

Step 1: Identify the negative exponent

The given expression is $$x^{-4}y^7$$. We can see the negative exponent in the term $$x^{-4}$$. We need to rewrite this term with a positive exponent.

Step 2: Rewrite the term with the negative exponent

Using the property of exponents: $$x^{-n}=\frac{1}{x^n}$$. We can rewrite the term with the negative exponent as follows: $$ x^{-4} = \frac{1}{x^4} $$

Step 3: Rewrite the entire expression

Now that we know $$x^{-4} = \frac{1}{x^4}$$, we can rewrite the given expression from the problem, like this: $$ x^{-4} y^{7} = \frac{1}{x^4} y^7 $$

Step 4: Write the final expression with only positive exponents

From Step 3, the expression with only positive exponents is: $$ \frac{1}{x^4} y^7 $$

Key Concepts

Negative Exponents

When we talk about negative exponents, we are referring to a specific rule in mathematics that helps simplify expressions. This rule states that a term with a negative exponent can be transformed into a positive one by taking the reciprocal of the base. In simpler terms, if you have a term such as \( x^{-n} \), it is the same as \( \frac{1}{x^n} \). This transformation is essential for expressing terms only in positive exponents, making calculations more straightforward and less error-prone.

This concept is particularly important in algebra where expressions can get complicated fast. Whenever you encounter a negative exponent, remember: you are essentially "flipping" the base. This core understanding can simplify handling even the most challenging algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These can range from simple expressions like \( 3x + 2 \) to more complex ones like \( x^{-4}y^{7} \). A key aspect of working with these expressions is being able to manipulate and simplify them, which often involves using exponent rules, especially when dealing with negative exponents.

Understanding how to rewrite algebraic expressions with positive exponents improves clarity and makes further mathematical operations like addition, subtraction, or even integration more manageable. When you simplify expressions as seen in our example \( x^{-4}y^{7} \), it gets converted to \( \frac{1}{x^4}y^7 \), which clears up any confusion that negative exponents might cause. In real-world applications, these expressions model situations or represent data, making it essential for accurate comprehension and calculations.

Exponent Rules

Mastering exponent rules is crucial for anyone studying algebra or more advanced mathematics. These rules dictate how to handle powers of numbers and variables, and they are fundamental for simplifying expressions. Let's list the key ones:

• Product of Powers: \( x^a \times x^b = x^{a+b} \).
• Power of a Power: \( (x^a)^b = x^{a \times b} \).
• Quotient of Powers: \( \frac{x^a}{x^b} = x^{a-b} \).
• Negative Exponent: \( x^{-n} = \frac{1}{x^n} \).
• Zero Exponent: \( x^0 = 1 \), provided that \( x eq 0 \).

Understanding these rules allows you to not only change negative exponents into positive ones, like in our exercise, but also tackle more complex algebraic operations. They provide a predictable framework when experimenting with expressions and functions, reinforcing both accuracy and efficiency in problem-solving.