Problem 55 - Algebra 1: Concepts and Skills

Question

Rewrite the expression with positive exponents. (Lesson 8.2) $$ \frac{d}{c^{-2}} $$

Step-by-Step Solution

Answer

The expression \( \frac{d}{c^{-2}} \) with positive exponents will be \( d * c^2 \).

Step 1: Identify the terms with negative exponents

The given expression is \( \frac{d}{c^{-2}} \). Here, \( c^{-2} \) has a negative exponent.

Step 2: Apply the power rule

According to the power rule, \( c^{-n} = \frac{1}{c^n} \). So, here, \( c^{-2} = \frac{1}{c^2} \).

Step 3: Rewrite the expression by replacing terms

Replacing \( c^{-2} \) in the given expression, we get \( \frac{d}{ \frac{1}{c^2} } \).

Step 4: Simplification

As \( \frac{a}{ \frac{b}{c} } = a * \frac{c}{b} \), we rewrite \( \frac{d}{ \frac{1}{c^2} } \) as \( d * c^2 \).

Key Concepts

Power Rule

Understanding the power rule is fundamental when working with expressions involving exponents. It's a simple yet powerful mathematical convention that can help simplify complex algebraic equations. In essence, the power rule provides a method to convert negative exponents into positive ones, and can be stated as follows: for any nonzero number a and any integer n, the expression with a negative exponent is given by \( a^{-n} = \frac{1}{a^{n}} \).

By applying the power rule, we can avoid dealing with negative exponents, which are often less intuitive. This comes in especially handy when solving algebraic problems. For example, in our exercise, we have \( c^{-2} \), which means that according to the power rule, we'd rewrite this as \( \frac{1}{c^{2}} \), thus converting the negative exponent into a positive one. This change not only simplifies the expression but also makes it easier to work with, whether we're adding, subtracting, multiplying, or dividing expressions with exponents.

Exponent Simplification

Exponent simplification involves reducing an expression with exponents to its simplest form. It often includes using the power rule for negative exponents and other rules like product of powers, quotient of powers, and power of a power. To clarify how simplification works, let's consider a fraction with an exponent in the denominator, similar to the exercise we are looking at.

Solving the exercise, we encounter a step where we rewrite \( \frac{d}{c^{-2}} \) as \( d * c^2 \). This is because when you divide by a fraction, it's equivalent to multiplying by its reciprocal. Here, simplifying the expression makes it more manageable and visibly more straightforward. Simplification is a crucial skill in algebra and higher-level math, as it allows one to streamline equations, making them easier to solve or manipulate for further operations.

Expressions with Exponents

Expressions with exponents often comprise various operations including multiplication, division, addition, and subtraction, combined with numbers raised to powers. An exponent dictates how many times a number, known as the base, is multiplied by itself. Positive exponents represent standard operations, but handling expressions that include them can sometimes be tricky and usually requires an understanding of exponent rules.

For instance, the expression \( d * c^2 \) is an example of an expression with a positive exponent. This expression indicates that number \( c \) is multiplied by itself once (since \( c^2 = c * c \)). It's crucial to recognize that when expressions include exponents, every operation can influence the simplification process. With positive exponents, the calculations are typically more straightforward. The key is to remember the exponent rules and apply them systematically, ensuring expressions are simplified and solutions are correctly obtained.