Problem 31 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ 6^{2} \cdot 6^{-4} $$

Step-by-Step Solution

Answer

The simplification of \(6^{2} \cdot 6^{-4}\) is \(1/36\).

Step 1: Understand the problem

The given expression is \(6^{2} \cdot 6^{-4}\). The goal is to simplify this expression without using a calculator.

Step 2: Apply the multiplication rule for exponents

To multiply powers with the same base, the rule is to add the exponents. Hence, \(6^{2} \cdot 6^{-4} = 6^{2 + (-4)} = 6^{-2}\).

Step 3: Evaluating 6 to the power of -2

The exponent -2 in \(6^{-2}\) means that the number has been inverted and squared. Thus \(6^{-2} = 1/(6^{2}) = 1/36\)

Key Concepts

Exponent Rules

When simplifying expressions that contain exponents, knowing the exponent rules is crucial for correctly evaluating or simplifying the problem. These rules allow us to perform operations on numbers with exponents efficiently. One fundamental rule is the product of powers rule, which states that when multiplying two expressions that have the same base, you can add the exponents. The mathematical expression for this rule is: for any non-zero number 'a', and any integers 'm' and 'n', \( a^m \cdot a^n = a^{(m+n)} \).

Using this rule, if we have an expression like \( 6^2 \cdot 6^{-4} \), we can add the exponents 2 and -4 to get \( 6^{-2} \). However, the rules don't stop there; other exponent rules include the power of a power rule, the quotient of powers rule, and the rule for dealing with zero as an exponent.

For instance, the power of a power rule says that when you have an exponent raised to another exponent, like \( (a^m)^n \), you multiply the exponents to simplify, giving us \( a^{mn} \). Understanding these rules makes working with exponents much more straightforward.

Negative Exponents

Negative exponents can be a source of confusion, but they follow a straightforward rule. A negative exponent indicates that the base is on the wrong side of the fraction line and it needs to be inverted. In simpler terms, for any non-zero number 'a' and any integer 'n', \( a^{-n} = 1/a^{n} \).

Therefore, when you encounter a term like \( 6^{-2} \), it does not mean 'negative six squared.' Rather, it signifies that the base (6 in this case) should be on the bottom of a fraction and to the power of the positive exponent (2). Thus, \( 6^{-2} \), following the negative exponent rule, becomes \( 1/6^{2} \), which is \( 1/36 \).

Mastering negative exponents is essential not only for simplifying algebraic expressions but also in various areas of mathematics, including calculus and scientific notation, where large and small values are often expressed with negative exponents for ease of computation and readability.

Evaluating Algebraic Expressions

Evaluating algebraic expressions is a fundamental skill in algebra. This process involves substituting numbers for variables and simplifying the expression using the order of operations and exponent rules. When an expression has exponents, it's important to accurately apply the rules of exponents as part of the evaluation.

For example, with the expression \( 6^{2} \cdot 6^{-4} \), you would first use the exponent rules to simplify the expression to \( 6^{-2} \), which means you’ve already started the evaluation process. After rewriting the expression using the negative exponents rule (\( 1/6^{2} \)), you then proceed to simplify the fraction by squaring the base, resulting in \( 1/36 \).

It’s crucial to work step-by-step, respecting the hierarchy of operations: addressing exponents, then multiplication or division, and finally addition or subtraction if present. By breaking down the process, evaluating complex algebraic expressions can become manageable and less intimidating for students.