Problem 51 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \left(c^{-1}\right)^{-4} $$

Step-by-Step Solution

Answer

Question: Rewrite the given expression without any negative exponents: $\left(c^{-1}\right)^{-4}$ Answer: $c^4$

Step 1: Rewriting the exponent

Using the rule of exponents, when we have an exponent raised to another exponent, we can multiply the exponents. So we have: $$ \left(c^{-1}\right)^{-4} = c^{-1 \times -4} $$

Step 2: Simplifying the expression

Now, we can multiply the exponent values: $$ c^{-1 \times -4} = c^{4} $$ The expression with only positive exponents is: $$ c^4 $$

Key Concepts

Positive exponents

Positive exponents are an important concept to understand when dealing with mathematical expressions. They indicate how many times a base number is multiplied by itself. For instance, in the expression \( c^4 \), the base \( c \) is multiplied by itself four times, which is \( c \times c \times c \times c \). Working with positive exponents is generally straightforward because they tell us to repeat multiplication.
To transform a negative exponent into a positive one, you can use the reciprocal. For example, \( c^{-n} \) becomes \( \frac{1}{c^n} \). This transformation ensures that expressions can be rewritten in terms of only positive exponents. This approach simplifies calculations and makes the expressions more intuitive to understand.

Simplifying expressions

Simplifying expressions with exponents involves making them as clear as possible, often by reducing negative exponents to positive ones. This is useful in algebra and simplifies the expressions for easier manipulation. For instance, given \( \left(c^{-1}\right)^{-4} \), we utilize the property that multiplying exponents leads us to \( c^{4} \).
To simplify such expressions, follow these steps:

• Perform operations within parentheses first. Here we see an exponent raised to another, so we apply rules of exponents.
• Apply the appropriate exponent laws by multiplying the exponents, as shown in \( \left(c^{-1}\right)^{-4} = c^{-1 \times -4} \).
• Simplify any resulting negative exponents by transforming them into positive ones to achieve \( c^4 \).

Simplification helps in revealing the structure of the expression, making it easier to solve or graph.

Exponent rules

Exponent rules are crucial guidelines that assist in the manipulation and simplification of expressions. They help in performing operations on exponents systematically and accurately. Let's explore the core exponent rules that apply to the problem:

• Product Rule: When multiplying like bases, add their exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient Rule: For dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power Rule: When an exponent is raised to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \). For \( \left(c^{-1}\right)^{-4} \), this rule turns the expression into \( c^4 \).
• Negative Exponent Rule: Converts negative exponents to positive ones by taking the reciprocal: \( a^{-n} = \frac{1}{a^n} \).

These rules enable one to manipulate exponential expressions to solve, simplify, or rearrange them without changing their value. Consistently applying these rules helps build a deeper understanding and fluency in algebra.