Question

Express the answers to these in scientific notation: (a) \(0.0095+\left(8.5 \times 10^{-3}\right)\) (b) \(653 \div\left(5.75 \times 10^{-8}\right)\) (c) \(850,000-\left(9.0 \times 10^{5}\right)\) (d) \(\left(3.6 \times 10^{-4}\right) \times\left(3.6 \times 10^{6}\right)\)

Short answer

The results in scientific notation are (a) \(1.8 \times 10^{-2}\), (b) \(1.135 \times 10^{11}\), (c) \(-5 \times 10^{4}\) , (d) \(1.296 \times 10^{3}\)

Step-by-step solution

Exercise (a): Addition

We need to add \(0.0095\) and \(8.5 \times 10^{-3}\). First, express \(0.0095\) also in scientific notation as \(9.5 \times 10^{-3}\). Then, add it to \(8.5 \times 10^{-3}\) to get \(18 \times 10^{-3}\) which is equivalent to \(1.8 \times 10^{-2}\) in scientific notation.

Exercise (b): Division

We need to divide \(653\) by \(5.75 \times 10^{-8}\). Convert \(653\) into scientific notation as \(6.53 \times 10^{2}\). Now, instead of divided it, multiply by the reciprocal, which gives us: \(6.53 \times 10^2 \times 1.73913043 \times 10^{8} = 1.135 \times 10^{11}\).

Exercise (c): Subtraction

We need to subtract \(9.0 \times 10^{5}\) from \(850,000\). Writing 850,000 in scientific notation gives us \(8.5 \times 10^{5}\). Then subtract \(9.0 \times 10^{5}\) to get \(-0.5 \times 10^{5}\), which is equivalent to \(-5 \times 10^{4}\) in scientific notation.

Exercise (d): Multiplication

We need to multiply \(\left(3.6 \times 10^{-4}\right)\) by \(\left(3.6 \times 10^{6}\right)\). Rule of multiplying scientific notations tells us to multiply the numbers and add the exponents, which gives us \(12.96 \times 10^{2}\), simplifying it gives \(1.296 \times 10^{3}\)

Key concepts

Scientific Notation Calculations

Scientific notation is a convenient way to express large or small numbers. It simplifies calculations, especially when dealing with extreme values. The format is: the sign, followed by a number between 1 and 10, and then an exponent of 10. For instance, the number 0.0003 is written as \(3 \times 10^{-4}\) in scientific notation.

In everyday science and engineering, scientific notation helps compare very different scales of values with ease. It's essential to learn how to carry out basic arithmetic operations—addition, subtraction, multiplication, and division—within this format, as these skills are widely applicable across various scientific fields.

Adding in Scientific Notation

To add numbers in scientific notation, first ensure the exponents of 10 are the same. For example, you can't directly add \(2.3 \times 10^{3}\) and \(5.4 \times 10^{2}\). You would need to adjust one of them to have matching exponents. Once they match, simply add the decimal parts and keep the exponent constant. If the result is not between 1 and 10, adjust the decimal part and exponent accordingly.

Remember, with scientific notation, clarity in representing the scale of numbers is just as important as getting the right answer. Always convert back into proper scientific notation if values exceed the 1-10 range.

Subtracting in Scientific Notation

Subtraction is similar to addition. Start by ensuring exponents are the same. If they're not, adjust one number accordingly. Then, subtract the decimal parts, keeping the exponent of 10 unchanged. Adjust if the result isn't between 1 and 10.

For instance, subtracting \(9.0 \times 10^{5}\) from \(8.5 \times 10^{5}\) may lead to negative scientific notation, such as \( -5 \times 10^{4}\), which signifies that the resulting quantity is smaller than the number from which we're subtracting.

Multiplying in Scientific Notation

When multiplying numbers in scientific notation, multiply the decimal parts and then add the exponents. The process is straightforward: for example, \(3.2 \times 10^{-3}\) multiplied by \(5 \times 10^{2}\) results in \(16 \times 10^{-1}\), which simplifies to \(1.6 \times 10^{0}\).

Unlike addition and subtraction, you don't need to have matching exponents to multiply. This process is easier but make sure the final result is also expressed in proper scientific notation.

Dividing in Scientific Notation

Division requires you to divide the decimal parts and subtract the exponent of the divisor from the exponent of the dividend. For example, \(8 \times 10^{5}\) divided by \(2 \times 10^{3}\) is \(4 \times 10^{2}\).

If you encounter a complex division, convert the divisor into its reciprocal and then multiply. This method is especially useful when dividing by numbers already written in scientific notation, as you can directly subtract the exponents.