Problem 40 - Algebra 1

Question

Evaluate the expression. $$x^{6}-1 \text { when } x=1.2$$

Step-by-Step Solution

Answer

The evaluation of the expression \(x^{6} - 1\) when \(x = 1.2\) is approximately 1.985984

Step 1: Substitute the value of x

The value of \(x\) is given as 1.2. Substitute \(x = 1.2\) in the expression \(x^6 - 1\). After substitution we get \(1.2^6 - 1\).

Step 2: Compute the expression

Now we want to compute the value of the expression \(1.2^6 - 1\). Using a calculator, \(1.2^6\) is approximately 2.985984. Subtract 1 from this number to get approximately 1.985984.

Key Concepts

Exponentiation

Exponentiation is a powerful mathematical operation. It involves raising a number, called the base, to the power of an exponent. This means multiplying the base by itself a certain number of times. For instance, in the expression \( x^6 \), \( x \) is the base and 6 is the exponent. This expression means that you multiply \( x \) by itself 6 times.

Let's consider a useful analogy: if you think of exponentiation as a series of repeated multiplications, it becomes easier. If we have \( x = 1.2 \), then \( 1.2^6 \) equals \( 1.2 \times 1.2 \times 1.2 \times 1.2 \times 1.2 \times 1.2 \).

• The base is the number you start with: \( 1.2 \).
• The exponent tells you how many times to use the base in a multiplication.

Using a calculator is often the simplest way to compute complex exponentiation, like finding the value of \( 1.2^6 \). Once you understand the basics, you can evaluate any other exponential expressions with ease.

Substitution Method

The substitution method in algebra is a technique used to solve expressions or equations. It involves replacing a variable with a given numerical value. This is often the first step in solving or simplifying algebraic expressions, as it helps in gaining insights about the expression's specific value.

In our exercise, you need to substitute \( x = 1.2 \) into the expression \( x^6 - 1 \). This turns the algebraic expression into a numeric calculation. Substitution helps in transforming variables into specific numbers and allows you to carry out subsequent calculations more easily.

• Start by identifying what value to substitute into the expression. Here it's \( x = 1.2 \).
• Replace every occurrence of the variable with this number to simplify the expression.

This method is crucial because it connects abstract algebraic expressions to concrete numbers, making the problem more tangible and easier to solve.

Mathematical Operations

Mathematical operations are fundamental to solving algebraic problems. They include addition, subtraction, multiplication, and division. In this exercise, you mostly deal with exponentiation and subtraction.

After you substitute and compute the exponentiation, you need to handle subtraction. In our example, once you find \( 1.2^6 \approx 2.985984 \), you subtract 1 from this value.

• Perform subtraction by aligning similar decimal places if doing manually.
• Using a calculator can help ensure precision when dealing with decimals.

This operation gives us the result \( 1.985984 \). Each mathematical operation plays a role in evaluating expressions, and understanding their order and application is essential in algebra.