Problem 20 - Elementary Algebra

Question

When simplifying the terms for the following problems, write each so that only positive exponents appear. $$ \left(c^{0}\right)^{-2}, \quad c \neq 0 $$

Step-by-Step Solution

Answer

Question: Simplify the expression $$\left(c^{0}\right)^{-2}$$ and rewrite it with only positive exponents. (Assume $$c \neq 0$$.) Answer: $$1$$

Step 1: Apply the exponent property of $$a^0 = 1$$

Since $$c^0 = 1$$ for any non-zero base $$c$$, the given expression can be rewritten as $$(1)^{-2}$$.

Step 2: Apply the exponent rule for negative exponents

Recall the exponent rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$. Use this rule to simplify $$(1)^{-2}$$: $$\frac{1}{(1)^2}$$.

Step 3: Simplify the expression

Now simplify the expression $$\frac{1}{(1)^2}$$. Since $$1^2 = 1$$, we get $$\frac{1}{1}$$. And finally, $$\frac{1}{1} = 1$$. So, the simplified expression with only positive exponents is $$1$$.

Key Concepts

Exponent Properties

Understanding the properties of exponents is crucial when simplifying algebraic expressions. Exponents, also known as powers, provide a concise way to represent repeated multiplication.

In the example, \(c^{0}\)^{-2}\, the base \(c\) has an exponent \(0\), and that entire expression is raised to the power of \(\-2\). A key property used here is the fact that any nonzero number raised to the power of zero is \(1\), often written as \(a^{0} = 1\), where \(a\) is any nonzero number.

This zero exponent rule is a shortcut to simplify expressions quickly, but it's only applicable when the base is not zero. In our exercise, the condition \(c \eq 0\) ensures that this rule is properly applied.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. Specifically, the rule states that \(a^{-n} = \frac{1}{a^n}\), where \(a\) is any nonzero number and \(n\) is a positive integer.

This property is essential when we want to express all terms with positive exponents. In the example, the term \(1^{-2}\) simplifies to \(\frac{1}{1^2}\) because of the negative exponent rule. Remember, the purpose of using negative exponents is not to generate negative numbers but to denote division by a number's power. Simplifying an expression to remove negative exponents often involves creating a fraction with the base in the denominator and the positive exponent.

Algebraic Simplification

Algebraic simplification encompasses the techniques used to rewrite expressions in simpler or more useful forms. Applying the rules of exponents is one such powerful simplification tool. It often makes solving equations, evaluating expressions, and understanding the structure of algebraic expressions much easier.

When we simplify expressions, we combine like terms, simplify fractions, and use exponent rules, among other operations, to make the expressions as clear and straightforward as possible. In our sample problem, the expression \(\frac{1}{1}\) simplifies to \(1\) after applying the properties of exponents.

Always aim to express your final answer with positive exponents and in the least complex form possible. Simplified expressions facilitate better comprehension and are less prone to errors in subsequent calculations.