Problem 48 - Algebra 2

Question

Use a calculator to approximate each value to three decimal places. $$ \sqrt[4]{(3500)^{2}} $$

Step-by-Step Solution

Answer

The approximate value is 187.791.

Step 1: Simplify the Expression Inside the Fourth Root

First, simplify the expression inside the fourth root: \((3500)^2\). This is calculated as:\[3500 \times 3500 = 12,250,000\]

Step 2: Apply the Fourth Root to the Simplified Expression

Now that we have simplified the expression inside the root, we need to find the fourth root of 12,250,000. The expression becomes:\[\sqrt[4]{12,250,000}\]

Step 3: Calculate the Fourth Root Using a Calculator

Use a calculator to approximate the fourth root of 12,250,000. Enter the expression and calculate:\[\sqrt[4]{12,250,000} \approx 187.791\]Ensure that the result is rounded to three decimal places.

Key Concepts

Roots and Radicals

Understanding roots and radicals is crucial when simplifying expressions in Algebra 2. A "root" is an operation that represents the inverse of taking a power. For example, the fourth root of a number undoes raising a number to the fourth power.
In simpler terms, if you need to find the fourth root of a number, you are looking for a value that, when raised to the fourth power, gives you the original number. Radicals are symbols used to denote roots, such as the square root, cube root, or fourth root, which usually look like the radical sign (√).

• Square root (√): The number that, when multiplied by itself, gives you the original number.
• Cube root (∛): The number that, when used three times in a multiplication, gives you the original number.
• Fourth root (∜): The number that, when used four times in multiplication, yields the original number.

In this exercise, taking the fourth root of 12,250,000 is handled in the same way. You are looking for a number, that when multiplied by itself four times, equals 12,250,000.

Exponents

Exponents are a way to represent numbers raised to a power. In other words, when you see a number with an exponent, it tells you how many times to multiply the number by itself. For example, the expression \((3500)^2\) means you multiply 3500 by 3500.
Exponents follow specific rules, which are helpful when simplifying or calculating values in algebraic expressions:

• Product of Powers Rule: When multiplying numbers with the same base, add the exponents.
• Power of a Power Rule: When raising an exponent to another power, multiply the exponents together.
• Zero Exponent Rule: Any number raised to the power of zero is one.

In our current problem, simply calculating \((3500)^2\) involves a straightforward multiplication, helpful for entering the expression into a calculator before taking the root.

Calculator Usage

In modern math, using a calculator is an essential skill, especially when dealing with large numbers or complex expressions that are cumbersome to calculate by hand. Calculators assist in getting precise values, exactly like the approximation to three decimal places required in this exercise.
For our task on finding the fourth root of 12,250,000, a calculator simplifies the process. Here's how you could approach using a calculator to find the value accurately:

• Input the number 12,250,000 into the calculator.
• Utilize the calculator's functionality to find the fourth root. This could be under a function menu or using a power button with fractional exponents, if applicable.
• Ensure you round your final result to three decimal places for precision, like how 187.791 is handled in this context.

Calculators come with many functions, so getting familiar with your calculator's specific operations can save time and improve accuracy, especially when working with more complex problems.