91Ó°ÊÓ

When facing exponential expressions like the one in our exercise, understanding negative exponents is crucial. A negative exponent represents the reciprocal of the base raised to the absolute value of that exponent. In simpler terms, to convert from a negative exponent to a positive, you take the reciprocal (flip) of the number. For example, with our expression \(3^{-3}\), you make it \(\frac{1}{3^3}\) or \(\frac{1}{27}\) as shown in the solution.

This rule is universal, meaning for any nonzero base \(b\) and negative exponent \(n\), the expression \(b^{-n}\) equals \(\frac{1}{b^n}\). Remember, this process doesn't change the value of the expression, it only presents it in a different form that is often easier to work with.

Practice with Negative Exponents

Try to convert the following expressions to see if you've grasped the concept: \(5^{-2}\), \(2^{-4}\), and \(\frac{1}{2^{-3}}\). Spoiler: You should end up with positive exponents throughout!

Once negative exponents are understood and we have a fraction, the next step is to simplify the fraction. Simplifying fractions means reducing them to their lowest terms, where the numerator and the denominator have no common factors other than 1. To simplify the fraction \(\frac{3}{27}\) in our exercise, you find the greatest common divisor (GCD) of the numerator and denominator. Here, both 3 and 27 are divisible by 3, so we divide them by 3 to get \(\frac{1}{9}\).

This process isn't just fundamental for evaluating algebraic expressions, but it's a critical skill for math in general. Simplifying can make numbers easier to understand and work with, especially when dealing with more complex algebraic or calculus problems.

Examples for Practice

To hone your skills, try simplifying these: \(\frac{8}{32}\), \(\frac{14}{28}\), and \(\frac{60}{75}\). Each has a common factor that can be divided out.

Our exercise involves not just numbers but an algebraic expression that includes an exponent. Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In this case, the expression \(3^{-3} \times 3\) is simplified through the understanding of negative exponents and arithmetic.

To deal with any algebraic expression effectively, it’s essential to follow the order of operations, known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction). This rule tells us to work out expressions in brackets first, then exponents, and so on, which allows for uniformity in solving expressions regardless of complexity.

Order of Operations Practice

For some extra practice, try evaluating these algebraic expressions following PEMDAS: \(4 \times (2+3)^2\) and \(\frac{1}{2} \times 6^2 - 3\).