Problem 72 - Calculus with Concepts in Calculus

Question

Modify the expression, and then find its approximate value by calculator or computer. $$ \frac{221^{907}}{221^{897}} $$

Step-by-Step Solution

Answer

The value of the expression is 1,445,551,875,539,456,376,221.

Step 1: Simplify the Expression

Simplify the given expression \( \frac{221^{907}}{221^{897}} \) by using the properties of exponents. According to the rule \( \frac{a^m}{a^n} = a^{m-n} \), we can rewrite the expression as follows: \[ \frac{221^{907}}{221^{897}} = 221^{907 - 897} = 221^{10} \].

Step 2: Calculate the Power

Now, calculate the power \( 221^{10} \). This requires using a calculator or a computer to find the value. The computation shows that:\[ 221^{10} = 1,445,551,875,539,456,376,221 \].

Key Concepts

Properties of Exponents

Exponents are a way to express repeated multiplication of the same number. Understanding the properties of exponents can greatly simplify the process of working with expressions involving powers.
One important property is the quotient of powers property, which states:

• \(\frac{a^m}{a^n} = a^{m-n}\)

This means when dividing two expressions with the same base, you simply subtract the exponents. This is exactly what was applied in the solution to the original exercise.
By rewriting \(\frac{221^{907}}{221^{897}}\) as \(221^{907-897}\), you see the simplification become \(221^{10}\). This property allows us to work with smaller, more manageable numbers.

Simplifying Exponents

Simplifying expressions with exponents involves rewriting them in a more compact or standard form. This often means reducing the expression using exponent rules like the product of powers, power of a power, or quotient of powers.
In the given expression \(\frac{221^{907}}{221^{897}}\), the quotient of powers property helped to simplify it to \(221^{10}\). This step significantly reduces the complexity, making it easier to evaluate.
The key is always to look for common bases and use the relationships between their exponents to simplify the calculations. This process is beneficial not only in reducing the numerical workload but also in understanding deeper algebraic concepts.

Power Calculation

After simplifying the expression, the next step is to calculate the resulting power, in this case, \(221^{10}\). Power calculation refers to determining the result of raising a number to an exponent.
Calculating high powers manually can be challenging, so using a calculator or computer software is practical. These tools handle the task with speed and accuracy, providing precise results.
The calculated value of \(221^{10}\) is 1,445,551,875,539,456,376,221, showcasing the immense size numbers can reach when exponentiated. Understanding how to manage large exponents helps in many areas of science and mathematics, making technology an indispensable aid in learning.