Problem 50 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 7 a^{2}(a-4)^{3} b^{-6} c^{-7} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression \(7 a^{2}(a-4)^{3} b^{-6} c^{-7}\) using only positive exponents. Answer: \(\frac{7 a^{2}(a-4)^{3}}{b^{6}c^{7}}\)

Step 1: Understand the given expression

The given expression is: $$ 7 a^{2}(a-4)^{3} b^{-6} c^{-7} $$ Our goal is to rewrite this expression using only positive exponents.

Step 2: Apply the property of exponents

Using the property \((\frac{a}{b})^n = \frac{a^n}{b^n}\), we can rewrite the expression as: $$ 7 a^{2}(a-4)^{3} \left(\frac{1}{b}\right)^{6} \left(\frac{1}{c}\right)^{7} $$

Step 3: Simplify the expression

Now, we can rewrite the expression as a single fraction by combining the numerators and denominators: $$ \frac{7 a^{2}(a-4)^{3}}{b^{6}c^{7}} $$

Step 4: Final Answer

The expression with only positive exponents is: $$ \frac{7 a^{2}(a-4)^{3}}{b^{6}c^{7}} $$

Key Concepts

Exponent Properties

Exponents are mathematical shorthand for expressing repeated multiplication of the same number by itself. It's crucial to understand various properties of exponents to simplify algebraic expressions effectively. One fundamental property is that positive exponents denote the number of times a base is multiplied by itself. Conversely, a negative exponent indicates division by the base multiplied by itself for the given number of times. To transform negative exponents into positive ones, you can apply the rule \( a^{-n} = \frac{1}{a^n} \), where \(a\) is the base, and \(n\) is the positive exponent. This property is particularly useful when you want to simplify expressions with negative exponents, as seen in the exercise. Another important exponent rule is the power of a product rule, which states that \( (ab)^n = a^n \times b^n \), allowing us to raise each factor inside a parenthesis to the power outside.

Simplifying Expressions

Simplifying algebraic expressions is a process of making them easier to read or solve, often by eliminating negative exponents and combining like terms. To simplify an expression, we apply various rules of algebra including the exponent properties already mentioned. After expressing all the exponents positively, we may need to perform additional steps to simplify the expression further. For example, if the expression contains fractions, we could look to combine and reduce them using common denominators. When dealing with polynomials, terms with the same variables and exponents should be combined. This process doesn't just make the expressions cleaner; it often makes the next steps of solving the equation or evaluating the expression much easier. In our example, simplification involved writing the entire expression as a single fraction, which is now ready for further operations if needed.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that together represent a specific value or set of values. They can include terms, which are the parts of an algebraic expression separated by addition or subtraction signs. The main goal when working with algebraic expressions is to manipulate them in ways that preserve their values while making them simpler or more suitable for solving. This often involves using exponent rules, distributing factors, factoring, and simplifying fractions, as we have seen in the provided example. A good understanding of how to manipulate algebraic expressions is essential for solving equations, understanding functions, and analyzing relationships between variables.