Question

Express the answers to the following calculations in scientific notation: (a) \(145.75+\left(2.3 \times 10^{-1}\right)\) (b) \(79,500 \div\left(2.5 \times 10^{2}\right)\) (c) \(\left(7.0 \times 10^{-3}\right)-\left(8.0 \times 10^{-4}\right)\) (d) \(\left(1.0 \times 10^{4}\right) \times\left(9.9 \times 10^{6}\right)\)

Short answer

(a) \(1.460 \times 10^2\); (b) \(3.18 \times 10^2\); (c) \(6.2 \times 10^{-3}\); (d) \(9.9 \times 10^{10}\).

Step-by-step solution

Add Numbers

Start with part (a): \(145.75 + (2.3 \times 10^{-1})\). First, convert \(2.3 \times 10^{-1}\) to a decimal which is 0.23.Then, add the two numbers: \(145.75 + 0.23 = 145.98\).Finally, express 145.98 in scientific notation: \(1.4598 \times 10^2\). Round to three significant figures: \(1.460 \times 10^2\).

Divide Numbers

For part (b): \(79,500 \div (2.5 \times 10^{2})\).Convert 79,500 to scientific notation: \(7.95 \times 10^4\).Perform the division: \((7.95 \times 10^4) \div (2.5 \times 10^2)\).Divide coefficients: \(7.95 \div 2.5 = 3.18\).Subtract exponents: \(10^4 \div 10^2 = 10^{4-2} = 10^2\).The result is \(3.18 \times 10^2\).

Subtract Numbers

Move on to part (c): \((7.0 \times 10^{-3}) - (8.0 \times 10^{-4})\).Convert both numbers to the same power of 10: \(7.0 \times 10^{-3} = 70 \times 10^{-4}\).Subtract: \(70 \times 10^{-4} - 8 \times 10^{-4} = 62 \times 10^{-4}\).Write the result in scientific notation: \(6.2 \times 10^{-3}\).

Multiply Numbers

For part (d): \((1.0 \times 10^4) \times (9.9 \times 10^6)\).Multiply coefficients: \(1.0 \times 9.9 = 9.9\).Add exponents: \(10^4 \times 10^6 = 10^{4+6} = 10^{10}\).The result is \(9.9 \times 10^{10}\).

Key concepts

addition and subtraction in scientific notation

When adding or subtracting numbers in scientific notation, the key is to make sure the exponents are the same. This way, it's like adding or subtracting regular numbers. Let’s break it down with an example from our exercise.

• In part (a), we added two numbers: 145.75 and 2.3 x 10^-1. First, we converted 2.3 x 10^-1 into 0.23. This gave us a decimal that matches the power of ten format.
• Then, 145.75 plus 0.23 equals 145.98. We then wrote this in scientific notation as 1.4598 x 10².
• Finally, rounding to three significant figures, we get 1.460 x 10².

For subtraction, like in part (c), ensure both numbers have the same exponent. Change 7.0 x 10^-3 to 70 x 10^-4 so you can subtract from 8.0 x 10^-4. It’s simple math afterward: 70 - 8 equals 62, resulting in 62 x 10^-4 or 6.2 x 10^-3 when formatted in scientific notation.

Consistency in exponents is crucial for easy calculation.

multiplication in scientific notation

Multiplying in scientific notation is straightforward because the coefficients and exponents can be handled separately. Let’s explore this using part (d) of the exercise.

• First, multiply the coefficients: In this case, 1.0 times 9.9 gives us 9.9.
• Next, you add the exponents: 10⁴ times 10⁶ becomes 10⁴⁺⁶ or 10¹⁰.
• The resultant expression is 9.9 x 10¹⁰.

By separating these steps, multiplication stays simple. No conversion is needed as both power and coefficient are manageable separately. This method reduces potential errors and keeps calculations straightforward while maintaining scientific notation efficiency.

It's often easier to check your work when it is structured like this.

division in scientific notation

For division involving scientific notation, you will divide the coefficients and subtract the exponents. Let’s consider part (b) to understand this process better.

• Begin by converting 79,500 into scientific notation, which is 7.95 x 10⁴.
• Now divide by 2.5 x 10².
• Divide the coefficients: 7.95 divided by 2.5 equals 3.18.
• Subtract the exponents: 10⁴ divided by 10² becomes 10^4-2 or 10².
• Your answer in scientific notation is 3.18 x 10².

This method makes it easier to handle large or small numbers by breaking down complex divisions into simpler steps. Subtracting exponents is a straightforward operation, and once the first step is done, the result is easily understood in scientific notation.

It’s a neat technique to avoid computational mishaps and keeps your results precise.

significant figures in scientific notation

Significant figures are important for indicating the precision of a calculation in scientific notation. They tell you which digits carry meaningful information and which do not.

• Consider part (a) again: After adding, we had 1.4598 x 10². By truncating to three significant figures, we get 1.460 x 10².
• This rounding results in potential error reduction when carrying numbers into further calculations.

Significant figures ensure precision in measurements and calculations by showing which numbers are reliable. They help in understanding the level of certainty in your measurements.

It's vital to consider significant figures whenever you work with scientific notation, especially in the context of scientific and engineering problems where accuracy is essential.