Problem 40 - Elementary Algebra

Question

Write the expressions for the following problems using only positive exponents. $$ \frac{4 c^{-2}}{b^{-6}} $$

Step-by-Step Solution

Answer

Question: Rewrite the expression $\frac{4 c^{-2}}{b^{-6}}$ using only positive exponents. Answer: $4b^6$

Step 1: Identify the negative exponents

In the given expression, we have two negative exponents: $$ c^{-2} \text{ and } b^{-6} $$ We need to rewrite these terms using only positive exponents.

Step 2: Apply the property to eliminate negative exponents

Using the property mentioned in the analysis section, we will rewrite the terms with negative exponents as follows: $$ c^{-2} = \frac{1}{c^2} $$ and $$ b^{-6} = \frac{1}{b^6} $$

Step 3: Substitute the new terms into the expression

Now, let's substitute these new terms into the given expression: $$ \frac{4 c^{-2}}{b^{-6}} = \frac{4 \frac{1}{c^2}}{\frac{1}{b^6}} $$

Step 4: Simplify the expression

To simplify this expression, we can multiply both the numerator and the denominator by the product of the denominators, in this case, \(c^2 b^6\). Doing so, we get: $$ \frac{4 \frac{1}{c^2}}{\frac{1}{b^6}} \cdot \frac{c^2 b^6}{c^2 b^6} = \frac{4c^2b^6}{c^2} $$

Step 5: Cancel out the common terms

Finally, we can cancel out the common terms in the numerator and the denominator: $$ \frac{4c^2b^6}{c^2} = 4b^6 $$ The final expression with only positive exponents is: $$ 4b^6 $$

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like addition, subtraction, multiplication, and division. They are often used to represent real-world situations mathematically. Understanding how to manipulate these expressions is crucial for solving equations and simplifying complex mathematical scenarios.

• Components: Variables, coefficients, constants, and operators make up an algebraic expression. For instance, in the expression \(3x + 5\), \(x\) is the variable, \(3\) is the coefficient, and \(5\) is the constant.
• Usage: Algebraic expressions provide a way to generalize mathematical operations and can be used to describe patterns, shapes, and changes over time.

To work with algebraic expressions, you often perform operations like addition, subtraction, and distribution of terms. Mastering these skills allows you to simplify expressions, which is essential when dealing with problems involving exponents.

Negative Exponents

Negative exponents denote division or reciprocals of a particular base raised to a positive exponent. In simple terms, a negative exponent tells you how many times to divide the base by itself. For instance, \(a^{-n} = \frac{1}{a^n}\).

• Application: Negative exponents appear frequently in algebraic expressions and are essential when analyzing patterns, decay, or inverse relationships.
• Example: Consider \(x^{-3}\), which can be rewritten as \(\frac{1}{x^3}\). This shows how a negative exponent simplifies to a reciprocal with a positive exponent.

You'll often transform negative exponents to positive ones to make calculations easier and expressions more understandable. This process is vital in simplifying complex algebraic expressions.

Exponent Rules

Exponent rules are tools that mathematicians use to manipulate expressions where numbers or variables are raised to powers. Understanding these rules allows you to break down complex exponentials into more manageable parts. Whether you're multiplying or dividing powers, these rules provide the framework for simplifying expressions.

• Product Rule: When multiplying two powers with the same base, you add the exponents: \(a^m \cdot a^n = a^{m+n}\).
• Quotient Rule: When dividing two powers with the same base, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \cdot n}\).

In the exercises, these rules are essential. For example, when converting negative exponents into positive ones, you apply these rules to simplify expressions. Each rule helps in handling different situations, ensuring expressions are precise and concise.