Problem 8 - Algebra 1

Question

Rewrite as an expression with positive exponents. $$3 c^{-5}$$

Step-by-Step Solution

Answer

The expression \(3c^{-5}\) when rewritten with positive exponents will be \(\frac{3}{c^5}\).

Step 1: Understand the power rule

One can rewrite any number or variable with a negative power by taking its reciprocal. The formula is \[a^{-n} = \frac{1}{a^n}\].

Step 2: Apply the power rule

Applying the power rule to the expression \(3c^{-5}\), only \(c^{-5}\) has a negative power. Thus we apply the rule to it resulting to \[\frac{1}{c^5}\].

Step 3: Rewrite and simplify the expression

The expression \(3c^{-5}\) can therefore be rewritten as \[3 \cdot \frac{1}{c^5} = \frac{3}{c^5}\].

Key Concepts

Understanding the Power Rule

The power rule is a handy tool in algebra that helps simplify expressions involving exponents. It's straightforward: any term with a negative exponent can be rewritten as the reciprocal of that term with a positive exponent. The basic formula for this is: \(a^{-n} = \frac{1}{a^n}\). This means, rather than dealing with negative exponents, which can be tricky, we use the power rule to transform it into something more manageable. For example, if you have \(x^{-3}\), using the power rule, you can rewrite it as \(\frac{1}{x^3}\). This makes further calculations easier to handle, as working with positive exponents is often simpler.

What is a Reciprocal?

A reciprocal is essentially what you get when you flip a number or a fraction upside down. In other words, the reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). If you have a whole number like 4, its reciprocal is \(\frac{1}{4}\). Understanding reciprocals is crucial when dealing with negative exponents, as we've seen with the power rule. For instance, the expression \(c^{-5}\) is equivalent to the reciprocal of \(c^5\), or \(\frac{1}{c^5}\).

• The reciprocal is straightforward: just invert the fraction or expression.
• This concept makes negative exponents easier to work with.

By recognizing the reciprocal of an expression, you avoid confusion, especially when simplifying complex algebraic expressions.

Simplifying Expressions

Simplifying expressions is the process of making a math expression as simple as possible. This often involves combining like terms, reducing fractions, and eliminating negative exponents by using the power rule and reciprocals. As seen in the example \(3c^{-5}\), we used the power rule to change \(c^{-5}\) into \(\frac{1}{c^5}\). This turned the expression from \(3c^{-5}\) into \(3 \cdot \frac{1}{c^5}\). After that, we simplify further into \(\frac{3}{c^5}\).

• Simplifying makes expressions easier to work with in equations and real-world applications.
• It reduces potential errors by creating expressions that are more straightforward to interpret and solve.

Take your time to simplify fully, ensuring all calculations are correct. It saves time in the long run and leads to a clearer understanding of underlying math concepts.