Question

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

Short answer

The short answers for each problem are as follows: a. \(3.125 \times 10^3\) b. \(10^6\) c. \(1\) d. \(1.818 \times 10^{-5}\) e. \(10^7\) f. \(10^6\) g. \(1 \times 10^{-7}\) h. \(10\)

Step-by-step solution

a. \(1 / 0.00032 \)

To find this result, divide 1 by 0.00032. We get: \[ \frac{1}{0.00032} = 3125 \] Now, we need to convert 3125 into scientific notation: \[ 3125 = 3.125 \times 10^3 \]

b. \(10^{3} / 10^{-3}\)

Use the properties of exponents to subtract the exponents: \[10^{3} / 10^{-3} = 10^{3-(-3)} = 10^6\]

c. \(10^{3} / 10^{3}\)

Use the properties of exponents to subtract the exponents: \[10^{3} / 10^{3} = 10^{3-3} = 10^0\] Since any number raised to the power of 0 is 1, the answer is: \[10^0 = 1\]

d. \(1 / 55,000\)

Divide 1 by 55,000 and then convert the result to scientific notation: \[ \frac{1}{55,000} = 1.8181818... \times 10^{-5} \]

e. \(\frac{(10^{5})(10^{4})(10^{-4})}{(10^{-2})}\)

Use the properties of exponents to simplify the expression: \[\frac{(10^{5})(10^{4})(10^{-4})}{(10^{-2})} = \frac{(10^{5+4-4})}{10^{-2}} = \frac{10^5}{10^{-2}} = 10^{5-(-2)} = 10^7\]

f. \(\frac{43.2}{(4.32 \times 10^{-5})}\)

Divide the numbers first and then handle the exponents: \[\frac{43.2}{(4.32 \times 10^{-5})} = \frac{10}{1 \times 10^{-5}} = \frac{10}{10^{-5}} = 10^{1-(-5)} = 10^6\]

g. \(\frac{(4.32 \times 10^{-5})}{432}\)

Divide the numbers first and then handle the exponents: \[\frac{(4.32 \times 10^{-5})}{432} = \frac{0.01 \times 10^{-5}}{1} = 0.01 \times 10^{-5} = 1 \times 10^{-7}\]

h. \(1 /(10^{5})(10^{-6})\)

Use the properties of exponents to simplify the expression: \[1 /(10^{5})(10^{-6}) = 1 /10^{5-6}= 1 /10^{-1}\] Now convert it into scientific notation: \[1 /10^{-1} = 1 \times 10^1 = 10\]

Key concepts

Exponents and Powers

Understanding exponents and powers is essential when dealing with scientific notation. Exponents, also known as indices or powers, represent the number of times a base number is multiplied by itself. The notation for exponentiation is a small number known as the exponent, written above and to the right of the base number. For example, in the expression \( 10^3 \), the base number is 10, and the exponent is 3, which means you must multiply 10 by itself twice, resulting in \( 10 \times 10 \times 10 = 1000 \).

When dividing numbers with the same base, one of the rules of exponents is to subtract the exponents. If the base is 10 as in \( 10^3 / 10^{-3} \), we simply subtract the exponent of the denominator from the exponent of the numerator to get \( 10^{3-(-3)} = 10^6 \). Another important rule to remember is that any nonzero number raised to the power of 0 is equal to 1, which is why \( 10^0 = 1 \).

One common mistake is to disregard the rules of exponents when mixed with coefficients. When combining expressions like \( 43.2 / (4.32 \times 10^{-5}) \), it's crucial to divide the coefficients first and then apply the exponent rule separately, which leads to \( 10^6 \) in this case.

Scientific Notation Standard Form

Scientific notation is a method to write very large or very small numbers in a more manageable form. The standard form of scientific notation requires expressing numbers as a product of two parts: a coefficient and a power of ten. The coefficient must be a number greater than or equal to 1 and less than 10, and it is followed by \( \times 10 \) raised to an exponent. For example, the number 3125 is written as \( 3.125 \times 10^3 \) in scientific notation.

To convert a number to scientific notation, one must first move the decimal point to create a new number from 1 up to 10; so, if we convert a small number like \( 0.00032 \) to scientific notation, we move the decimal four places to the right which gives us 3.2, and because we moved the decimal to the right, we use a negative exponent resulting in \( 3.2 \times 10^{-4} \). Conversely, if we move the decimal point to the left for very big numbers, the exponent will be positive.

Mathematical Operations with Exponents

When performing mathematical operations with exponents, there are key rules that help to simplify calculations. For multiplication, when you multiply two exponents with the same base, you add the exponents. If we take the expression \( (10^5)(10^4)(10^{-4})/(10^{-2}) \), the exponent rules allow us to combine the bases by adding and subtracting their exponents, leading to \( 10^7 \) as the simplified form.

When dividing with exponents, as mentioned earlier, you subtract the exponent of the divisor from the exponent of the dividend, like in the expression \( 4.32 \times 10^{-5} / 432 \), turning into \( 1 \times 10^{-7} \). In addition to multiplication and division, raising a power to another power requires you to multiply the exponents, and taking the root is similar to raising to a fractional exponent. This set of rules is vital for performing mathematical operations efficiently within scientific notation and beyond.