Problem 22 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for the indicated value of \(x\).\(4 x^{-3} \quad x=2\)

Step-by-Step Solution

Answer

The evaluated expression is 0.5.

Step 1: Understand the expression

The expression we have is \(4 x^{-3}\). The exponent of \(x\) is \(-3\) which means \(x^{-3} = \frac{1}{x^{3}}\). So our expression can be rewritten as \(4 \cdot \frac{1}{x^{3}}\).

Step 2: Substitute the value

Substitute \(x = 2\) into our rewritten expression. Thus the expression becomes \(4 \cdot \frac{1}{2^{3}}\).

Step 3: Simplify the expression

Simplify the expression by performing the operations. Calculate the denominator \(2^{3} = 8\) and then divide 4 by this result: \(4 \cdot \frac{1}{8} = 0.5\).

Key Concepts

Understanding Exponents

When working with algebraic expressions, it is important to understand the role of exponents. An exponent is a small number written above and to the right of a base number. It tells us how many times this base number is multiplied by itself. For example, in the expression \(x^{-3}\), the exponent is \(-3\). A negative exponent indicates the reciprocal of the base raised to the positive of the exponent. Therefore, \(x^{-3} = \frac{1}{x^{3}}\). In simpler terms, a negative exponent transforms the expression into a fraction, where the base is the divisor.

• Positive exponents: represent repeated multiplication.
• Negative exponents: indicate reciprocal multiplication.
• Understanding exponents is key to transforming expressions properly.

Recognizing the inverse relationship in negative exponents helps in converting them into expressions that are easier to manipulate.

Applying Substitution

Substitution is a technique used to evaluate algebraic expressions by replacing variables with given numbers. In this exercise, we replace \(x\) with \(2\). By doing so, we transform the expression from a variable statement into a numerical one that can be easily computed. For the expression \(4 x^{-3}\), substituting \(x = 2\) gives us \(4 \cdot \frac{1}{2^{3}}\). Always remember to double-check that you're substituting all instances of the variable in the expression.

• Identify the value to substitute for the variable.
• Replace the variable with the given number in the entire expression.
• Ensure you maintain the correct form as you substitute.

Substitution helps to evaluate expressions concretely, allowing you to move onto simplification with actual numbers.

Process of Simplifying Expressions

Simplifying expressions involves performing mathematical operations to reduce an expression to its simplest form. After substitution, you may need to simplify by following standard arithmetic operations. In our exercise, after substituting \(x = 2\) into \(4 \cdot \frac{1}{2^{3}}\), we simplify by executing the operation of exponentiation first, where \(2^{3} = 8\).Next, multiply 4 by the fraction \(\frac{1}{8}\), which simplifies to \(0.5\). Simplification involves breaking down the operations step-by-step to ensure accuracy.

• Finish all operations within the expression.
• Simplify fractions by finding the equivalent simplest form.
• Check your result for correctness as you simplify.

Simplifying expressions correctly ensures that you have an accurate answer, easy to read and interpret. This final step highlights the importance of carefully following the order of operations (PEMDAS/BODMAS) in mathematics.