Problem 44 - Elementary and Intermediate Algebra

Question

Simplify. Do not use negative exponents in the answer. \(\frac{6}{k^{-2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(6k^2\).

Step 1: Review Negative Exponent Rule

The rule for negative exponents states that \(a^{-n} = \frac{1}{a^n}\) and vice versa: \(\frac{1}{a^{-n}} = a^n\). This means that moving the base with a negative exponent from the denominator to the numerator will change the exponent's sign.

Step 2: Apply the Negative Exponent Rule

Given \(\frac{6}{k^{-2}}\), we apply the rule \(\frac{1}{k^{-2}} = k^2\), resulting in \(6 \times k^2\).

Step 3: Simplify the Expression

The expression \(6 \times k^2\) simplifies to \(6k^2\). This is the final simplified form, with no negative exponents.

Key Concepts

Exponent Rules

Understanding exponent rules is key to mastering algebra and simplifying expressions effectively. Exponents denote how many times a number, known as the base, is multiplied by itself. When we work with exponents, knowing a few key rules can simplify calculations greatly.

The most commonly used rules for exponents include:

• Product of Powers: When multiplying two expressions with the same base, add the exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When dividing two expressions with the same base, subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a Power: To raise a power to another power, multiply the exponents: \((a^m)^n = a^{m \times n}\).
• Negative Exponent Rule: A negative exponent indicates reciprocal: \(a^{-n} = \frac{1}{a^n}\).
• Zero Exponent Rule: Any base raised to the zero power is 1: \(a^0 = 1\), where \(a eq 0\).

Understanding these rules, especially the negative exponent rule, is crucial for solving algebra problems that involve moving bases between numerators and denominators efficiently.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent a particular value or set of values. These expressions form the foundation of algebra and are used extensively to model real-world problems.

Components of algebraic expressions often include:

• Constants: Fixed numbers without variables, like 6 in the expression \(6k^2\).
• Variables: Symbols like \(k\) that represent unknown or changeable values.
• Coefficients: Numbers that are multiplied by variables, such as the 6 in \(6k^2\), indicating 6 times the variable \(k^2\).
• Operators: Mathematical symbols (e.g., +, -, \(\times\), \(\div\)) used to perform operations in the expression.

To work with algebraic expressions, one needs to understand how to combine like terms, use distributive properties, and apply exponent rules. This understanding allows for simplification, solving equations, and performing substitutions as necessary in calculations.

Simplification of Expressions

Simplifying expressions is an essential skill in algebra, helping to make complex problems more manageable. Simplification means reducing an expression to its simplest form where no further reduction of terms is possible. This often involves combining like terms, reducing fractions, and applying exponent rules efficiently.

Key steps in simplifying include:

• Remove Negative Exponents: Apply the rule \(a^{-n} = \frac{1}{a^n}\) to eliminate negative exponents. For instance, in the given exercise, turning \(\frac{6}{k^{-2}}\) into \(6 \times k^2\).
• Combine Like Terms: Sum or subtract terms that have the same variables and powers. This step ensures the expression is as concise as possible.
• Reduce Fractions: Simplify fractions within the expression to their lowest terms, if applicable.

By practicing simplification, solving algebra problems becomes more straightforward, and you'll gain the ability to see patterns in expressions easily. This is why mastering the principles of simplification can make a huge difference in becoming proficient in algebra.