Question

Express the answers to the following calculations 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

(a) \(1.8 \times 10^{-2}\); (b) \(1.14 \times 10^{10}\); (c) \(-5.0 \times 10^{4}\); (d) \(1.296 \times 10^{3}\).

Step-by-step solution

Calculate and Convert to Scientific Notation (a)

First, calculate the sum of the numbers: \(0.0095 + 8.5 \times 10^{-3} = 0.0095 + 0.0085 = 0.018\). Next, convert \(0.018\) to scientific notation. To do this, move the decimal point two places to the right to get \(1.8 \times 10^{-2}\).

Calculate and Convert to Scientific Notation (b)

First, divide the numbers: \(653 \div 5.75 \times 10^{-8} = 113565.217391304348\). Then, convert this number to scientific notation. This is approximately \(1.14 \times 10^{10}\) after moving the decimal 10 places to the left.

Calculate and Convert to Scientific Notation (c)

First, subtract the numbers: \(850,000 - 9.0 \times 10^{5} = 850,000 - 900,000 = -50,000\). Convert \(-50,000\) to scientific notation by moving the decimal point four places to the left: \(-5.0 \times 10^{4}\).

Calculate and Convert to Scientific Notation (d)

First, multiply the numbers: \((3.6 \times 10^{-4}) \times (3.6 \times 10^{6}) = 12.96 \times 10^{2}\). Finally, convert \(12.96 \times 10^{2}\) to scientific notation: \(1.296 \times 10^{3}\).

Key concepts

Addition and Subtraction in Scientific Notation

Adding and subtracting in scientific notation can seem tricky at first, but it's all about aligning the exponents. In order to add or subtract numbers that are both in scientific notation, the exponents must be the same. If they aren't equal, you must adjust one of the numbers so that the exponents match.
Imagine you're adding apples, you can't add them together properly if you're counting in different units! For example:

• Convert both numbers to the same exponent, like adjusting both to the scale of \(10^{-2}\), if needed.
• Once the bases are the same, add or subtract the coefficients (the number part in front).
• Lastly, convert the result back to scientific notation, if it isn't already.

In the problem, the first step was completed by adjusting the decimal places to ensure both numbers were expressed in terms of the same power of ten before proceeding with the addition.

Multiplication and Division in Scientific Notation

Multiplying and dividing numbers in scientific notation is simpler, as you don't need to align the exponents. However, you must understand how to handle the exponents efficiently.
For multiplication:

• Multiply the coefficients (the numbers before the exponents).
• Add the exponents according to the laws of exponents: if the numbers are \(10^a\times10^b\), the result becomes \(10^{a+b}\).
• Ensure that your final product is in proper scientific notation.

For division:

• Divide the coefficients.
• Subtract the exponent of the denominator from the exponent of the numerator. So, \(10^a\div10^b = 10^{a-b}\).
• Again, make sure your answer follows scientific notation convention.

This method was applied correctly in the division example from the exercise, transforming the calculation into a form that's easier to interpret and manage.

Decimal to Scientific Notation Conversion

Converting a decimal to scientific notation is like organizing a messy backpack. You want to tidy everything into a simplified form. To convert:

• Identify where the decimal point should move so that there's exactly one non-zero digit to its left.
• Count the number of places you moved the decimal. This becomes the power of ten. Move right and it's negative, left and it's positive.
• Express the number as a number between 1 and 10 times \(10\) raised to the identified power.

In the exercise, moving the decimal to standardize the form involved shifting decimal points to express the answer neatly in scientific notation, illustrating how powerful this mathematical tool is when organizing numbers, large or small.

Significant Figures in Scientific Notation

Significant figures are crucial when expressing precision in scientific notation. They tell us how many digits in a number are meaningful, reflecting the precision of the measurement or calculation.
You must remember:

• All non-zero numbers are significant.
• Any zeros between significant figures are also significant.
• Leading zeros are not significant, but trailing zeros in a decimal portion are.

In scientific context, the student's focus should be on maintaining consistency in significant figures through calculations, ensuring results reflect the initial data's accuracy. The multiplication example outcome in the exercise carefully reflects the impact, showing how trickily yet carefully, significant digits must be preserved throughout.