Problem 59 - Chemistry Principles and Practice

Question

A Calculate the result of the following equation, and use the convention of significant figures to express the answer correctly. $$ x=\frac{10^{121}}{10^{-121}} \times 1.01 $$

Step-by-Step Solution

Answer

The result is \( 1.01 \times 10^{242} \) with three significant figures.

Step 1: Solve the division of powers

When dividing powers with the same base, subtract the exponents. Here, \( x = \frac{10^{121}}{10^{-121}} \). So, you perform the subtraction of exponents: \[ x = 10^{121 - (-121)} = 10^{121 + 121} = 10^{242} \]

Step 2: Multiply by the decimal factor

Now multiply \( x = 10^{242} \) by 1.01. Thus, perform the multiplication: \[ x = 10^{242} \times 1.01 = 1.01 \times 10^{242} \]

Step 3: Determine the number of significant figures

The decimal factor 1.01 has three significant figures. Therefore, the final result should also be expressed with three significant figures for consistency.

Step 4: Express the final result

Given that the number of significant figures is set by 1.01, the final result is: \[ x = 1.01 \times 10^{242} \] This represents the answer in scientific notation with the correct number of significant figures.

Key Concepts

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. It involves writing numbers as a product of a decimal and a power of ten. For example, instead of writing out 1,000,000, you can use scientific notation to express it as \(1 \times 10^6\). This format makes it easier to handle, read, and perform calculations with extremely large or tiny values.
When writing numbers in scientific notation, you typically adjust the decimal point so that it appears after the first non-zero digit. The exponent indicates how many places the decimal point has been moved to convert the original number into the standardized scientific notation form. This approach simplifies arithmetic operations such as multiplication and division with these numbers.

Multiplication of Powers

Multiplying numbers with powers involves specific rules. When you multiply powers of ten, you add the exponents, which simplifies computations considerably. For example, if you have \(10^a \times 10^b\), the result is \(10^{a+b}\), because you are combining the effects of each ten's power.
This principle helps in reducing complex multiplications to simpler operations with exponents. For instance, when calculating \(10^{242} \times 1.01\), where 1.01 is not a power but a decimal value, you multiply the numerical coefficient (1.01) by the power separately. The result stays in the format where the numerals are combined, keeping the power of ten intact.

Division of Powers

Division of powers that share the same base also obeys a straightforward rule. When you divide \(10^a\) by \(10^b\), you subtract the exponent \(b\) from \(a\), resulting in \(10^{a-b}\). This keeps calculations simpler, especially when working with scientific notation.
Using the example from the problem, dividing \(10^{121}\) by \(10^{-121}\) translates to \(10^{121-(-121)} = 10^{242}\). This changes a cumbersome division into a neat subtraction of exponents, streamlining the entire calculation process.

Exponents

Exponents are an essential mathematical shorthand used to denote repeated multiplication of a base number. For example, \(10^3\) means 10 multiplied by itself three times (\(10 \times 10 \times 10\)).
Exponents make handling large computations manageable, as they allow for large mathematical transformations with simple arithmetic operations. Through rules like adding and subtracting exponents when multiplying or dividing powers with the same base, you can simplify calculations that would otherwise be lengthy and complex.
Mastering the use of exponents and the rules that govern them is crucial for understanding scientific notation and efficiently performing calculations involving large numbers.