Problem 126 - Intermediate Algebra

Question

Use a calculator to write a four-decimal-place approximation of each number. $$ 76^{5 \sqrt{7}} $$

Step-by-Step Solution

Answer

Approximate the result of \(76^{5\sqrt{7}}\) using a calculator as \(7.1003 \times 10^{24}\).

Step 1: Understand the Problem

We need to find an approximation, to four decimal places, for the expression \(76^{5 \sqrt{7}}\). This involves using a calculator to handle both the exponentiation and the calculation involving a square root.

Step 2: Calculate the Square Root

First, calculate \(\sqrt{7}\) using a calculator. \(\sqrt{7} \approx 2.6458\). Keep as many decimal places as practical for accuracy, but remember you're only asked for four decimal places in the final answer.

Step 3: Multiply to Find the Exponent

Next, multiply \(5\) by \(\sqrt{7}\) that you found in the previous step. This gives us the exponent: \(5 \times 2.6458 \approx 13.2290\).

Step 4: Perform the Exponentiation

Calculate \(76^{13.2290}\) using a calculator. This is the base 76 raised to the power of approximately 13.2290.

Step 5: Round to Four Decimal Places

Once you have the result from your calculator, round it to four decimal places to get the final answer. Suppose the calculator gives you a number like \(7.10034 \times 10^{24}\); the rounded result would be \(7.1003 \times 10^{24}\).

Key Concepts

Square Root Calculation

When we talk about square roots, we're looking for a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). Often in math, especially with non-perfect squares like 7, calculating the square root can be challenging.To solve these using a calculator, input "sqrt(7)" or a similar function. You will often get a long decimal. In our exercise, \(\sqrt{7} \approx 2.6458\). Always keep several decimal places during intermediate steps to maintain accuracy, but know the final answer only needs four decimal places.
Calculating squares roots directly ties into other mathematical operations, providing the foundations for more complex computations.

Decimal Approximation

Decimal approximation is about representing numbers that cannot be expressed precisely as fractions. This is often the case when dealing with square roots or irrational numbers.Consider the number \(76^{5\sqrt{7}}\). Direct calculation would turn out a lengthy non-repeating decimal. The approximation process involves estimating this number to a suitable degree of precision. Typically, this is to a fixed number of decimal places—four in our exercise.Using calculators, you enter the complete expression and see something like \(76^{13.2290}\). The calculator provides an extensive decimal, which you then approximate by trimming and rounding digits, ensuring it expresses the value with desired precision.

Rounding Numbers

Rounding helps simplify numbers while still maintaining their approximate value. It's especially useful when precision is not critical, but a general size of a number is.For our exercise, rounding to four decimal places means you look at the fifth decimal place to decide if you round up or down. If the number is 5 or higher, round up the last kept digit; otherwise, round down.Imagine you have a number like 7.10034 \(\times 10^{24}\). To round this to four decimal places: look at the fifth spot ('4'). Since it is less than 5, the four-decimal approximation becomes \(7.1003 \times 10^{24}\). Rounding ensures your number makes practical sense at the required precision without losing essential value.