Problem 83 - Elementary Algebra

Question

Write the following expressions using only positive exponents. Assume all variables are nonzero. $$ \frac{6}{a^{2} b^{-4}} $$

Step-by-Step Solution

Answer

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

Step 1: Identify negative exponents

In order to rewrite the expression with only positive exponents, we need to identify the terms with negative exponents. In this case, we have: $$ b^{-4} $$

Step 2: Apply rule to change negative exponent to positive

We will now apply the rule that \(a^{-n} = \frac{1}{a^n}\) to the \(b^{-4}\) term in order to change its exponent from negative to positive. This results in the following transformation: $$ b^{-4} = \frac{1}{b^4} $$

Step 3: Rewrite the expression with positive exponents

Now that we have changed the negative exponent to a positive exponent, we can rewrite the original expression by substituting the new term for the original \(b^{-4}\) term. We get: $$ \frac{6}{a^{2} b^{-4}} = \frac{6}{a^2 \cdot \frac{1}{b^4}} $$

Step 4: Simplify the expression

Finally, we will simplify the expression by multiplying the numerators and denominators together: $$ \frac{6}{a^2 \cdot \frac{1}{b^4}} = \frac{6 \cdot 1}{a^2 \cdot b^4} = \frac{6}{a^2 b^4} $$ The rewritten expression with only positive exponents is: $$ \frac{6}{a^2 b^4} $$

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental aspect of mathematics that involve numbers, variables, and operators. In simple terms, they are combinations of letters and numbers linked together by arithmetic operations.
An algebraic expression can include:

• Constants: Numbers with a fixed value, like 6 in our example.
• Variables: Symbols representing numbers that can change, such as \(a\) and \(b\) in our expression.
• Operators: These include addition (+), subtraction (-), multiplication (⋅), and division (÷).

Algebraic expressions help us write down mathematical problems or scenarios in a concise form. They are essential for solving equations and inequalities in algebra.

Exponent Rules

Exponent rules are guidelines for simplifying expressions that involve powers of numbers or variables. When dealing with exponents, it's crucial to remember the basic rules:

• Product Rule: \(a^m \cdot a^n = a^{m+n}\) – Add exponents when multiplying with the same base.
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\) – Subtract exponents when dividing with the same base.
• Power Rule: \((a^m)^n = a^{m \cdot n}\) – Multiply exponents when raising a power to another power.
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\) – A negative exponent indicates the reciprocal of the base raised to the positive exponent.

In our example, we applied the negative exponent rule to rewrite \(b^{-4}\) as \(\frac{1}{b^4}\). Using these rules helps in restructuring algebraic expressions to a more workable form.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form while retaining their original values. The goals may include:

• Eliminating negative exponents.
• Combining like terms.
• Reducing fractions to their simplest form.
• Factoring, when applicable.

In the given expression \(\frac{6}{a^{2} b^{-4}}\), we sought to eliminate the negative exponent by applying the relevant rule. Once we changed \(b^{-4}\) to \(\frac{1}{b^4}\), we multiplied the fraction's denominators to achieve a simplified expression \(\frac{6}{a^2 b^4}\). This process often requires careful application of exponent rules and algebraic manipulation to achieve a clean, understandable result.