Question

Write each expression with positive exponents only. $$a b^{-5} c^{-2}$$

Short answer

The expression with positive exponents only is \(\frac{a}{b^5 c^2}\).

Step-by-step solution

Understanding Negative Exponents

Inspect the given expression, which contains negative exponents. Recall from algebra that any term with a negative exponent can be written as the reciprocal of the term with a positive exponent. In other words, for any nonzero number 'a' and any integer 'n', the rule is that 'a^{-n} = 1/a^n'.

Rewrite with Positive Exponents

Apply the rule to each term with a negative exponent. For the given expression, we'll move both 'b^{-5}' and 'c^{-2}' to the denominator and change the exponents to positive. The expression becomes 'a * 1/b^5 * 1/c^2', which simplifies to 'a/(b^5 c^2)'.

Key concepts

Exponent Rules

When dealing with exponents, it's critical to understand the rules that govern how to manipulate them. One of the most important principles is the treatment of negative exponents. The rule states that any number raised to a negative exponent can be converted into a positive exponent by taking its reciprocal. For example, if you have a term like \(a^{-n}\), you can rewrite it as \(\frac{1}{a^n}\). This is invaluable when simplifying expressions, as it allows you to turn an expression with negative exponents into one that only involves positive exponents, which are often easier to work with.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. These expressions can sometimes include exponents, which are shorthand notations for repeated multiplication of the same factor. As exponents appear frequently in algebra, knowing how to manipulate them is crucial for simplifying algebraic expressions effectively. When expressions have negative exponents, as in our exercise, converting these to positive exponents helps remove any potential confusion and prepares the expression for further operations like addition, subtraction, multiplication, or division.

Simplifying Expressions

Simplifying algebraic expressions often involves several steps: combining like terms, using the distributive property, and applying exponent rules, among others. When you encounter negative exponents, the simplification process includes transforming those exponents into positive ones to achieve a standardized form. In our example \(ab^{-5}c^{-2}\), simplification leads to \(\frac{a}{b^5c^2}\). The absence of negative exponents means it follows the convention of writing expressions with positive exponents only, which is generally preferable in mathematical communication.

Reciprocals

The concept of a reciprocal is fundamental when dealing with negative exponents. The reciprocal of a number is simply 1 divided by that number. For instance, the reciprocal of 2 is \(\frac{1}{2}\). In the context of negative exponents, writing the term with a negative exponent in the denominator of a fraction as a positive exponent is its reciprocal. This is why in our exercise, the terms with negative exponents are moved to the denominator and given positive exponents, thereby utilizing the reciprocal to transform the expression into a form with positive exponents only.