Question

Rewrite as an expression with positive exponents. $$m^{-2}$$

Short answer

\(1/{m^2}\)

Step-by-step solution

Applying the rule of exponents

In the given expression, the negative exponent can be rewritten by taking its reciprocal. So by applying the rule \(a^{-n} = 1/{a^n}\), the given expression \(m^{-2}\) can be rewritten as \(1/{m^2}\).

Key concepts

Exponent Rules

Understanding exponent rules is crucial for simplifying algebraic expressions and solving equations involving powers. One of the most important rules is dealing with negative exponents. According to the basic exponent rules, any number or variable raised to a negative exponent can be converted to its reciprocal with a positive exponent. For example, if you have an expression like \(x^{-n}\), this can be rewritten as \(\frac{1}{x^n}\).

In our exercise, we used this rule to transform \(m^{-2}\) into a form with only positive exponents. By definition, \(m^{-2}\) is equivalent to \(\frac{1}{m^2}\) because the negative exponent dictates that we take the reciprocal of the base (which is \(m\)) and then raise it to the positive of that exponent (which is 2 in this case).

Remember, this rule holds true regardless of whether the base is a number, a variable, or a more complex expression. It's a foundational concept that frequently appears in algebra, and mastering it opens doors to solving a wide range of mathematical problems.

Reciprocal of a Number

The reciprocal of a number is simply switching the numerator and denominator of that number when it is written in fraction form. In other words, the reciprocal of any number \(a\) (when \(a\) is not zero) is \(\frac{1}{a}\). This concept extends to variables as well. If you have a variable \(b\), its reciprocal is \(\frac{1}{b}\).

Understanding reciprocals becomes particularly important when dealing with negative exponents. When we are given \(m^{-2}\), we can find its reciprocal by flipping the base and exponent to get \(\frac{1}{m^2}\), as discussed previously. This is a practical application of the reciprocal in algebraic terms.

Being comfortable with finding reciprocals can also assist in division problems involving fractions, since dividing by a number is the same as multiplying by its reciprocal. Moreover, reciprocals are utilized in advanced mathematics, including solving equations involving ratios, rates, and proportional reasoning.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, and division) without an equal sign. These expressions can represent a variety of relationship types and are a cornerstone of algebra.

When working with algebraic expressions, it is essential to know how to manipulate and simplify them using the rules of algebra. This includes using exponent rules to work with powers and employing the concept of reciprocals to simplify expressions with negative exponents, such as the transition from \(m^{-2}\) to \(\frac{1}{m^2}\) that we've covered.

Algebraic expressions with exponents often appear in polynomials, rational expressions, and equations. By mastering how to handle them, you can solve an extensive array of problems. Using the principles of algebra, you learn to reorder and combine terms effectively, factor expressions, work with inequalities, and eventually solve complex equations that model real-world situations.