Problem 51 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ 18 b^{-6}\left(b^{2}-3\right)^{-5} c^{-4} d^{5} e^{-1} $$

Step-by-Step Solution

Answer

Question: Express the given expression using only positive exponents: $$18 b^{-6}\left(b^{2}-3\right)^{-5} c^{-4} d^{5} e^{-1}$$. Answer: $$\frac{18 d^5}{b^6 (b^2 - 3)^5 c^4 e}$$.

Step 1: Identify Negative Exponents

First, let's identify the negative exponents in the given expression $$18 b^{-6}\left(b^{2}-3\right)^{-5} c^{-4} d^{5} e^{-1}$$. There are four negative exponents: $$b^{-6}, \left(b^2 - 3\right)^{-5}, c^{-4}, e^{-1}$$.

Step 2: Rewrite the Negative Exponents as Positive Exponents

We can rewrite negative exponents as positive exponents by taking the reciprocal of the base. For negative exponent terms: $$b^{-6} = \frac{1}{b^6}$$ $$\left(b^2 - 3\right)^{-5} = \frac{1}{(b^2 - 3)^5}$$ $$c^{-4} = \frac{1}{c^4}$$ $$e^{-1} = \frac{1}{e}$$ Now, substitute these positive exponent terms back into the expression: $$18 \cdot \frac{1}{b^6} \cdot \frac{1}{(b^2 - 3)^5} \cdot \frac{1}{c^4} \cdot d^5 \cdot \frac{1}{e}$$

Step 3: Combine the Fractions and Simplify

Now, we will combine the fractions and simplify the expression: $$\frac{18 \cdot d^5}{b^6 \cdot (b^2 - 3)^5 \cdot c^4 \cdot e}$$ This expression is now written using only positive exponents. The final answer is: $$\frac{18 d^5}{b^6 (b^2 - 3)^5 c^4 e}$$.

Key Concepts

Negative Exponents

Negative exponents may initially seem tricky, but they're actually quite simple to handle once you understand the basic rule. The rule for negative exponents is as follows: when a base is raised to a negative exponent, you take the reciprocal of the base and switch the sign of the exponent to positive.
For example, consider the expression with a negative exponent: \(x^{-n}\). You can rewrite this as \(\frac{1}{x^n}\). This means you switch from multiplication to division while converting the exponent to positive. It's like flipping the fraction around the base.
This rule applies to any expression with one or more negative exponents. In our original expression, each term with a negative exponent is rewritten as a fraction to make the exponent positive.

Exponent Rules

Understanding exponent rules helps simplify and manage expressions effectively. These rules are quite straightforward and knowing them can make algebra a lot easier.
**Key Exponent Rules Include:**

• **Product of Powers Rule:** When multiplying two expressions with the same base, add their exponents. For instance, \(a^m \cdot a^n = a^{m+n}\).
• **Quotient of Powers Rule:** When dividing two expressions with the same base, subtract the exponents. For example, \(\frac{a^m}{a^n} = a^{m-n}\).
• **Power of a Power Rule:** To raise a power expression to another exponent, multiply the exponents: \((a^m)^n = a^{mn}\).
• **Negative Exponent Rule:** Covered above, this rule involves taking the reciprocal to convert the exponent to positive \(a^{-n} = \frac{1}{a^n}\).

These rules are invaluable for simplifying and reworking expressions into more manageable formats, like rewriting negative exponents as positive ones.

Fraction Simplification

Simplifying fractions is crucial in algebra, especially when dealing with complex expressions. In essence, simplification means reducing fractions to their simplest form.
If you have a complex fraction, the goal is to simplify it by combining like terms and reducing the expression to make it easier to understand. For instance, if you encounter a fraction like \(\frac{numerator}{denominator}\), look for any numbers or variables common to both the numerator and the denominator.

• You can simplify by dividing both by their greatest common factor.
• Sometimes, you can cancel out terms if they appear identically in both the numerator and the denominator.
• Ensure to maintain the integrity of the expression by only canceling or simplifying when it's mathematically valid.

This process streamlines the expression, leading to a cleaner result as shown in the final simplified fraction with positive exponents.

Algebraic Expressions

Algebraic expressions are a way of representing numbers, operations, and other symbols in a structured format. They can include variables, numbers, and operators.
For example, an expression like \(18b^{-6}(b^2-3)^{-5}c^{-4}d^5e^{-1}\) encompasses multiple components - numbers (18), variables (b, c, d, e), and operations (multiplication and power).

• **Understanding Variables:** Variables are symbols (like x, y, z) used to represent numbers. They're placeholders for values that are not yet known.
• **Managing Complexity:** Algebraic expressions can be simplified or rearranged using rules like those of exponents or by simplifying fractions.
• **Converting Terms:** Make expressions easier to work with by converting negative exponent terms using the reciprocal method, as in the exercise.

Successfully manipulating algebraic expressions allows you to solve equations or simplify complex problems, enhancing your math skills.