Problem 105 - Algebra A Combined Function

Question

Use a calculator to write a four-decimal-place approximation of each number. $$ 8^{1 / 4} $$

Step-by-Step Solution

Answer

The four-decimal-place approximation of \(8^{1/4}\) is 1.6818.

Step 1: Understand the expression

We're asked to approximate \(8^{1/4}\), which means we're looking for the fourth root of 8. In exponential form, this is expressed as raising 8 to the power of \(1/4\).

Step 2: Use a calculator to find the approximation

Input \(8^{0.25}\) into your calculator to compute the fourth root of 8. Ensure that your calculator is set to provide results with at least four decimal places.

Step 3: Record the approximation

Upon calculation, you should get the result \(8^{1/4} \approx 1.6818\). This value is the four-decimal-place approximation for the expression.

Key Concepts

Exponential Expressions

Exponential expressions are mathematical constructs in which numbers are raised to the power of an exponent. They are a way of representing repeated multiplication concisely. For example, in the expression \(8^{1/4}\), 8 is the base and \(1/4\) is the exponent.

Understanding exponents involves knowing that:

• Positive integer exponents (e.g., \(8^3\)) represent repeated multiplication. Here, 8 is multiplied by itself three times.
• Fractional exponents, like \(8^{1/4}\), are used to denote roots. The expression \(8^{1/4}\) means finding the fourth root of 8.
• Negative exponents, such as \(8^{-1}\) involve reciprocals, meaning \(8^{-1} = \frac{1}{8}\).

When working with exponential expressions, recognizing these different forms helps in understanding the type of operation required. Particularly, fractional exponents are a powerful method to express roots, which leads us to the concept of calculating roots.

Decimal Approximation

Decimal approximation is a method of expressing numbers that can't be represented exactly, in a form that is easy to understand and use. Calculators are often utilized to simplify this process by allowing us to convert complex numbers into a specific number of decimal places.

To achieve decimal approximation:

• Input the expression into a calculator. In our example, inputting \(8^{0.25}\) lets the calculator handle the complex calculations involved.
• Ensure your calculator is set to display enough decimal places. For a four-decimal-place approximation, adjust settings to show at least four digits after the decimal point.
• Interpret the result. The output will be a decimal that approximates the exact value. Thus, \(8^{1/4} \approx 1.6818\).

Decimal approximation is foundational in making complex computations tangible, providing a practical means of using irrational numbers day-to-day.

Fourth Root Calculation

Calculating the fourth root of a number involves finding a value which, when raised to the power of four, equals the original number. It's mathematically represented in exponential terms with a fractional exponent \(\frac{1}{4}\).

Follow these steps for calculating fourth roots:

• Recognize that \(x^{1/4}\) is equivalent to calculating the fourth root of \(x\).
• Enter \(x^{0.25}\) into a calculator, since \(0.25\) is the decimal representation of \(\frac{1}{4}\).
• Interpret the calculator's result. For example, entering \(8^{0.25}\) yields approximately \(1.6818\), meaning that this is the numerical approximation for which raised to the fourth power becomes 8.

Understanding root calculations is essential for interpreting complex mathematical relationships, and calculators make this often intricate task much more manageable.