Problem 34 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ 4^{-2} a^{-3} b^{-4} c^{5} $$

Step-by-Step Solution

Answer

Answer: \(\dfrac{c^5}{16a^3b^4}\)

Step 1: Rewrite each term with a negative exponent using the property \(a^{-n} = \dfrac{1}{a^n}\)

First, we'll rewrite each term that has a negative exponent using only positive exponents. $$ 4^{-2} a^{-3} b^{-4} c^{5} = \dfrac{1}{4^2} \cdot \dfrac{1}{a^3} \cdot \dfrac{1}{b^4} \cdot c^5 $$

Step 2: Simplify the expression

Next, we can simplify the expression by multiplying the fractions and separating the numerators and denominators: $$ = \dfrac{1 \cdot 1 \cdot 1 \cdot c^5}{4^2 \cdot a^3 \cdot b^4} = \dfrac{c^5}{4^2 a^3 b^4} $$

Step 3: Calculate \(4^2\) and write the final expression

Now, we can calculate \(4^2\), which is 16. Then, write the final expression using only positive exponents: $$ = \dfrac{c^5}{16a^3b^4} $$ The given expression \(4^{-2} a^{-3} b^{-4} c^{5}\) has been rewritten using only positive exponents as \(\dfrac{c^5}{16a^3b^4}\).

Key Concepts

Exponent Rules

Exponent rules are essential tools in algebra that help us work with powers and learn how to manipulate them effectively. One important rule is the negative exponent rule. A negative exponent means that the base is on the wrong side of a fraction.

• The rule states that any base with a negative exponent can be flipped to the other side of a fraction, making the exponent positive. For example, \(a^{-n} = \frac{1}{a^n}\).
• This rule allows us to convert expressions with negative exponents into a more standard form with positive exponents.

When working with expressions, converting all negative exponents to positive ones simplifies the computation and makes the expression easier to manage. This concept is not only useful for computation but also helps in aligning with standard mathematical conventions.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations. They can look complex, especially when they involve exponents, but understanding their components makes them manageable.

• Every term in an algebraic expression can have exponents attached to variables which indicate repeated multiplication of the base.
• Exponents show how many times a number, called the base, is multiplied by itself.

In our exercise, the expression \(4^{-2} a^{-3} b^{-4} c^{5}\) combines several variables and coefficients raised to both negative and positive powers. By applying exponent rules, we can rearrange this expression so all exponents are positive, making the task of simplifying much easier. Understanding how to handle algebraic expressions with exponents is fundamental in solving more complex equations and algebraic problems.

Simplifying Expressions

Simplifying expressions involves applying mathematical rules to reduce expressions to their simplest form. This process makes the expressions easier to work with.

• Simplifying involves rewriting terms using algebraic principles, such as converting negative exponents to positive ones as shown above.
• For the expression \(\frac{c^5}{4^2 a^3 b^4}\), every element is already in its simplest form after using the previous steps.
• The numerator has only one term, \(c^5\), which requires no additional simplification.

By simplifying expressions, we ensure that we use the easiest and most understandable forms in calculations, which is particularly useful for solving equations and comparing mathematical statements. Furthermore, understanding simplification leads to better problem-solving and analytical skills.