Question

Use your calculator to evaluate these expressions. Express the final answer in proper scientific notation. a) \(\left(9.09 \times 10^{8}\right) \div\left[\left(6.33 \times 10^{9}\right) \times\left(4.066 \times 10^{-7}\right)\right]=?\) b) \(9,345 \times 34.866 \div 0.00665=?\)

Short answer

a) \(3.534 \times 10^{5}\); b) \(4.90 \times 10^{7}\).

Step-by-step solution

Understand the Expression in Part (a)

Evaluate the given expression part by part. First, identify the operations: division and multiplication. The expression is given as \(\left(9.09 \times 10^{8}\right) \div\left[\left(6.33 \times 10^{9}\right) \times\left(4.066 \times 10^{-7}\right)\right]\). This means you first need to evaluate the expression inside the brackets.

Calculate the Expression Inside the Brackets in Part (a)

Multiply \(6.33 \times 10^{9}\) by \(4.066 \times 10^{-7}\). To do this, multiply the coefficients (6.33 and 4.066) and add the exponents of 10. \(6.33 \times 4.066 \approx 25.71878\), and \(10^{9} \times 10^{-7} = 10^{2}\). Thus the result is \(25.71878 \times 10^{2}\) or \(2.571878 \times 10^{3}\) after adjusting for proper scientific notation.

Divide the Original Numerator by Result from Step 2 in Part (a)

Now divide \(9.09 \times 10^{8}\) by \(2.571878 \times 10^{3}\). Divide the coefficients: \(9.09 \div 2.571878 \approx 3.534\). Subtract the exponents: \(10^{8} \div 10^{3} = 10^{5}\). The final answer is \(3.534 \times 10^{5}\).

Understand the Expression in Part (b)

The expression given for part (b) is \(9,345 \times 34.866 \div 0.00665\). Approach this problem by first multiplying and then dividing as per the order of operations.

Multiply in Part (b)

Multiply 9,345 by 34.866 to get \(9,345 \times 34.866 \approx 325,787.49\).

Divide to Find the Final Answer in Part (b)

Now divide the result from the previous step, 325,787.49, by 0.00665. Doing so gives \(325,787.49 \div 0.00665 \approx 48,993,609.7744\). Express this in scientific notation as \(4.899360977 \times 10^{7}\), or approximately \(4.90 \times 10^{7}\) for simplicity.

Key concepts

Division

In mathematics, division is an important operation that involves dividing one number by another. In our original exercise, division is part of setting up the expression. When you divide numbers in scientific notation, you need to take care of both the base numbers (the coefficients) and the powers of ten separately:

• Coefficients: Divide the numerical values.
• Exponents: Subtract the exponent of the divisor from the exponent of the dividend.

For example, in part (a), we divided the coefficient of 9.09 by the result of the multiplication from the brackets. Alongside, we subtracted the exponents, which changed the powers of ten.
This method keeps the structure of scientific notation, making it easy to handle very large or very small numbers.

Multiplication

Multiplication in scientific notation is usually straightforward. It involves multiplying both the numbers' coefficients and adding their exponents.
For instance, consider the multiplication inside the brackets in part (a) of the exercise:

• First, multiply the coefficients: 6.33 and 4.066, yielding approximately 25.71878.
• Then, add the exponents of ten: 9 and -7, which result in 2. Hence, the power of ten in the product is now 10².

Doing this keeps the solutions in a neat, understandable scientific notation. When dealing with large numbers, breaking them into scientific notation simplifies multiplication since it combines simple arithmetic with basic operations on exponents.

Order of Operations

The order of operations is crucial in solving mathematical problems. It ensures that calculations are performed in a consistent manner. When dealing with complex expressions, like the ones in our exercise, knowing the correct sequence is key.
Typically, the rules dictate:

• Parentheses: First, solve any expressions inside brackets.
• Exponents: Solve exponents after taking care of what's inside the brackets.
• Multiplication and Division: These operations are solved from left to right as they appear. In our exercise, part (a) involved first multiplying within the brackets, then dividing the result.
• Addition and Subtraction: These come last, but they weren't needed in this specific task.

Following these steps ensures the correctness of the solution and helps avoid common mistakes.

Exponents

Exponents are a shorthand way to represent repeated multiplication of a number by itself. They are a key component of scientific notation, which is used to simplify calculations with very large or very small numbers.
In our exercise, exponents helped simplify the multiplication and division of powers of ten:

• Multiplication: When multiplying numbers with the same base, add the exponents. For instance, in part (a), multiplying 10⁹ by 10^-7 resulted in 10².
• Division: Conversely, when dividing numbers with the same base, subtract the exponents. In part (a), dividing 10⁸ by 10³ resulted in 10⁵.

These simple rules underpin the efficiency of scientific notation, allowing easier handling of numbers across vast scales.