Problem 46 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 4 x^{-6} y^{2} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression with positive exponents only: \(4 x^{-6} y^2\) Answer: \(\frac{4}{x^6} y^2\)

Step 1: Identifying the negative exponent in the expression

In the given expression, \(4 x^{-6} y^2\), we can see the term \(x^{-6}\) has a negative exponent.

Step 2: Rewriting the expression using the exponent rule

Now we can rewrite the term with the negative exponent as its reciprocal with a positive exponent: \(x^{-6} = \frac{1}{x^6}\).

Step 3: Substitute back into the expression

Replace \(x^{-6}\) in the expression with the new term: $$4 x^{-6} y^2 = 4 \cdot \frac{1}{x^6} y^2$$.

Step 4: Simplify the expression

Combine the coefficients: $$4 \cdot \frac{1}{x^6} y^2 = \frac{4}{x^6} y^2$$.

Step 5: Final Answer

The expression with only positive exponents is: $$\frac{4}{x^6} y^2$$.

Key Concepts

Understanding Negative Exponents

Negative exponents might seem challenging at first, but they are quite simple once you get the hang of them. A negative exponent indicates that the base of the exponent should be reciprocated and turned into a fraction. This means if you have a term like \(x^{-n}\), it is equivalent to \(\frac{1}{x^n}\). The negative sign merely tells us to "flip" the base to its reciprocal, changing the exponent to be positive.
For example:

• \(x^{-3} = \frac{1}{x^3}\)
• \(y^{-1} = \frac{1}{y}\)

This concept is essential when simplifying expressions because it helps convert all exponents in the expression to positive, making calculations easier and more comprehensible.

Navigating Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and exponents that represent a quantity. They often include terms that can be simplified or combined using various algebraic rules. In our initial expression \(4x^{-6}y^2\), the numbers 4, \(x^{-6}\), and \(y^2\) are the components of the algebraic expression.

• Constant Term: This is the number that stands alone; in this case, the number 4.
• Variable Term: These are the terms containing variables with exponents, such as \(x^{-6}\) and \(y^2\).

Understanding how an expression is constructed is critical to simplify it or rewrite it in a particular way, such as using only positive exponents.

Mastering Exponent Rules

Exponent rules provide the groundwork for simplifying expressions and understanding how to manipulate powers in mathematics. They tell us what to do when multiplying or dividing like terms with exponents:

• Product of Powers: When multiplying two exponents with the same base, add their exponents. For instance, \(a^m \cdot a^n = a^{m+n}\).
• Quotient of Powers: When dividing two exponents with the same base, subtract their exponents, such as \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: Raise a power to another power by multiplying the exponents: \((a^m)^n = a^{mn}\).
• Negative Exponents: As discussed, they dictate using reciprocals. \(a^{-m} = \frac{1}{a^m}\).

These rules are the foundation for solving and simplifying expressions, clearly demonstrating how to operate with exponents efficiently.

Simplifying Expressions with Exponents

Simplifying expressions is a crucial skill in algebra that involves rewriting expressions in their most straightforward form. This often means using the rules of exponents to combine or reduce terms. When simplifying expressions, aim for clarity and ease of interpretation.

• Convert negative exponents to positive using reciprocal values.
• Combine like terms where applicable, such as adding or subtracting constants or variable terms with the same base and exponent.

In our example, we began with \(4x^{-6}y^2\). By recognizing the negative exponent, we used our exponent rules to rewrite it as \(\frac{4}{x^6}y^2\), which is simpler and uses only positive exponents.
Simplifying expressions helps make complex algebraic problems more manageable and easier to solve.