Problem 23 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for the indicated value of \(x\).\(7 x^{-2} \quad x=4\)

Step-by-Step Solution

Answer

The evaluated expression is \(7 / 16\).

Step 1: Understanding the given expression

The given algebraic expression has an exponent of -2. This implies reciprocal, meaning that \(x^-2\) actually becomes \(1/(x^2)\). So the expression \(7 x^{-2}\) can be rewritten as \(7 / (x^2)\).

Step 2: Substitution

Now substitute the value of \(x\) (which is 4) into the expression. This gives us \(7 / (4^2)\)

Step 3: Evaluation

Now, square the denominator (i.e., \(4^2\) is 16), making the expression \(7 / 16\)

Key Concepts

Exponent Notation

Exponent notation is a way of expressing repeated multiplication of the same factor. Instead of writing a number multiple times, we use a base and an exponent to simplify notation. For example, the exponential expression \( 4^2 \) conveys that we are multiplying 4 by itself once: \( 4 \times 4 \). In our exercise, the exponent is a negative integer \( -2 \), which is less common but very useful.

Negative exponents indicate a reciprocal relationship. Essentially, \( x^{-n} \) is equivalent to \( 1/(x^n) \). Therefore, if we have \( x^{-2} \) as in the exercise, this means we have \( 1/(x^2) \). This notation helps us work with very large or very small numbers efficiently and is crucial in algebra, science, and engineering.

Substitution in Algebra

we replace \(x\) and compute the new expression, which leads us to the next concept, simplifying expressions.

Simplifying Expressions

Simplifying expressions means to reduce an algebraic expression to its simplest form. It involves combining like terms, performing arithmetic operations, and reducing fractions when possible. This step is essential for making expressions easier to understand and less prone to errors when calculating.

Looking at the substitution result \(7 / (4^2)\), our next step is to simplify. We calculate the exponent \(4^2\) to get 16. Now our expression reads \(7 / 16\), which is already in its simplest form because 7 and 16 have no common factors other than 1. Simplification is a skill that makes algebra cleaner and more efficient, paving the way for solving more complex problems.