Write each of the following numbers in standard scientific...

Write each of the following numbers in standard scientific notation. a. $\frac{1}{0.00032}$ b. $\frac{10^{3}}{10^{-3}}$ c. $\frac{10^{3}}{10^{3}}$ d. $\frac{1}{55,000}$ e. $\frac{\left(10^{5}\right)\left(10^{4}\right)\left(10^{-4}\right)}{\left(10^{-2}\right)}$ f. $\frac{43.2}{\left(4.32 \times 10^{-5}\right)}$ g. $\frac{\left(4.32 \times 10^{-5}\right)}{432}$ h. $\frac{1}{\left(10^{5}\right)\left(10^{-6}\right)}$

Write each of the following numbers in standard scientific notation.
a. $\frac{1}{0.00032}$
b. $\frac{10^{3}}{10^{-3}}$
c. $\frac{10^{3}}{10^{3}}$
d. $\frac{1}{55,000}$
e. $\frac{\left(10^{5}\right)\left(10^{4}\right)\left(10^{-4}\right)}{\left(10^{-2}\right)}$
f. $\frac{43.2}{\left(4.32 \times 10^{-5}\right)}$
g. $\frac{\left(4.32 \times 10^{-5}\right)}{432}$
h. $\frac{1}{\left(10^{5}\right)\left(10^{-6}\right)}$

Key Concepts

Scientific Notation

Scientific notation is a method of writing very large or very small numbers as the product of a number (the coefficient) between 1 and 10 and a power of 10. This format simplifies calculations and comparison of numbers with many digits or many zeros.

Exponent Rules

Exponent rules refer to the mathematical laws that govern operations on numbers expressed as powers of 10. For example, when multiplying numbers in scientific notation, you add the exponents; when dividing, you subtract them. These rules are essential for simplifying expressions in scientific notation.

Standard Form

Standard form in the context of scientific notation means expressing a number such that the coefficient is at least 1 and less than 10, with the value adjusted by an appropriate power of 10. This convention ensures consistency and ease of interpretation.

Decimal Conversion

Decimal conversion involves rewriting a decimal number into scientific notation by moving the decimal point to create a new coefficient that fits the standard form, and then multiplying by the corresponding power of 10 to preserve the original value.

Transcript

00:01 Today we will be rewriting each of the below numbers in scientific notation.

00:06 Let's start with a.

00:08 Now this is a fraction, so we should convert the bottom and the top to scientific notation before we combine the two.

00:15 Now one is simply one, we don't need to do anything on the top.

00:18 For the bottom, however, we need to move the decimal point back 1, 2, 3, 3 times in order to get 3 .2 times 10 to the negative 4.

00:30 As the original number was less than 1.

00:33 Now 1 over 3 .2 is equal to 0 .3125.

00:41 And now this 0 .3125 is multiplied by 10 to the 4 because if you remember, something divided by 10 to the negative of whatever number, in this case 1 over 10 to the negative 4, is also equal to 10 to the positive whatever that original number was.

01:00 So we know that this is our answer.

01:03 However, we are not done because standard scientific notation has two parts.

01:09 The part where 10 to the something is multiplied by another part, we have 10 to the 4.

01:14 However, the final part, which is a number between 1 and 10, is what we do not have yet, as this number is less than 1.

01:22 Now we have an easy fix to this.

01:23 All we have to do is move the decimal point back by 1.

01:27 And so our final answer is 3 .125 times, because we move the decimal point back once, it is now no longer 10 to the 4, but 10 to 3.

01:38 So that's our final answer for a.

01:42 Now, let's move on to b.

01:45 Just an arrow here.

01:47 Now, b is 10 to the 3 over 10 to the negative 3.

01:51 This can also be written as 10 to the 3 times 10 to the 3.

01:55 And if we have something that is the same number but two different exponents, just add the exponents together.

02:01 So this is equal to 10 to 3 plus 3 in parentheses, which is essentially 1 times 10 to the 6th.

02:16 So that is our final answer for b.

02:19 Moving on to c, here we can see that this is the same number divided by itself.

02:25 10 to the 3 divided by 10 to the 3 is quite simply 1.

02:30 So we set this equal to 1.

02:32 If our final answer for c is just 1.

02:36 Moving on to d.

02:38 Again, like we do, part a, we're going to convert the bottom denominator into scientific notation first.

02:45 So that would be 0 .55 times 10 to the 5.

02:56 So we move it.

02:57 1, 2, 3, 4, 5.

03:01 Now, we divide 1 by 0 .55, which is equal to 1 .81 recurring...

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