Problem 18 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

Question: Rewrite the expression with a negative exponent to an equivalent expression using only a positive exponent: \(x^{-2}\). Answer: \(\frac{1}{x^2}\)

Step 1: Recall the property of negative exponents

Recall that a negative exponent can be rewritten as the reciprocal of the positive exponent. In other words, given a base and a negative exponent, the expression can be written as follows: $$ a^{-n} = \frac{1}{a^n} $$ Here, the base is \(x\) and the negative exponent is \(-2\). So, we will rewrite the given expression using the above property.

Step 2: Apply the property to the given expression

Rewriting the given expression using the property of negative exponents, we get: $$ x^{-2} = \frac{1}{x^2} $$ So, the given expression with only positive exponents is: $$ \frac{1}{x^2} $$

Key Concepts

Understanding the Reciprocal of Positive Exponents

When working with negative exponents, it's essential to recognize the underlying principle that guides their transformation into positive exponents. To rewrite an expression like \(x^{-2}\) using only positive exponents, we utilize the concept of reciprocals. A reciprocal is simply a flipped version of the number when thinking about fractions. For instance, the reciprocal of \(x\) is \(1/x\), and vise versa.

Using the reciprocal, we can convert a negative exponent into a positive one. The formula \(a^{-n} = 1/a^n\) makes this process clear. It indicates that to get rid of a negative exponent, we can write it as the reciprocal of the base raised to the positive exponent. Therefore, for our original expression \(x^{-2}\), the equivalent expression with a positive exponent is \(1/x^2\). This understanding of reciprocals is pivotal in both simplifying expressions and solving more complex algebraic equations.

Algebraic Expressions and Negative Exponents

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. One such operation includes exponents, which can sometimes be negative. The presence of a negative exponent in an algebraic expression signals the need for transformation to a positive exponent for the ease of further calculation or simplification.

The usage of negative exponents within algebraic expressions isn't just a stylistic choice; it's a functional part of mathematical language that conveys a specific meaning—namely, the multiplicative inverse or reciprocal. The process of converting a negative exponent into a positive one by writing it as the reciprocal of the base raised to the positive exponent does not change the value of the expression; instead, it simplifies it to a form that's often easier to work with, especially in the context of more complex algebraic manipulations like solving equations or simplifying fractions.

Exponent Properties and Simplifying Expressions

Exploiting exponent properties is a fundamental skill in algebra that helps in simplifying expressions and solving equations. One critical property is related to negative exponents, which we've discussed. But there are more exponent properties to be aware of:

• The Product of Powers Property: \(a^m \times a^n = a^{m+n}\) says that when multiplying two powers with the same base, you add the exponents.
• The Quotient of Powers Property: \(a^m / a^n = a^{m-n}\) tells us that when dividing two powers with the same base, you subtract the exponents.
• The Power of a Power Property: \((a^m)^n = a^{mn}\) indicates that when raising a power to a power, you multiply the exponents.
• And of course, the Power of a Product Property: \((ab)^n = a^n b^n\), which shows that when raising a product to a power, you raise each factor to the power separately.

Understanding these properties allows you to manipulate and simplify expressions involving exponents efficiently. For instance, seeing \(x^{-2}\) as the reciprocal \(1/x^2\) is essentially applying the Quotient of Powers Property, where you consider the expression as \(x^0/x^2\text{, and thus }\text{you subtract the exponents} (-2)\text{ from } 0\text{ to get } x^{0-(-2)}=x^2\). These properties are intertwined, and mastering them will greatly enhance your algebraic prowess.