Problem 10 - Introductory Chemistry Atoms First

Question

Convert \(4.68 \times 10^{-1}\) to standard notation.

Step-by-Step Solution

Answer

The given number in scientific notation, \(4.68 \times 10^{-1}\), is equal to 0.468 in standard notation.

Step 1: Understanding scientific notation

We are given a number in scientific notation as \(4.68 \times 10^{-1}\). In scientific notation, a number is expressed as the product of two factors - one is a number between 1 and 10 (in this case, 4.68), and the other is a power of 10 (in this case, \(10^{-1}\)).

Step 2: Convert the exponent of 10 to a decimal

The exponent, \(-1\), tells us that we need to move the decimal point in 4.68 one place to the left, as negative exponent means the actual number will be a fraction. Calculate the value for \(10^{-1}\): \[10^{-1} = \frac{1}{10^1} = \frac{1}{10}\] So, \(4.68 \times 10^{-1}\) becomes: \(4.68 \times \frac{1}{10}\)

Step 3: Multiply the decimal

Now multiply the two factors together: \(4.68 \times \frac{1}{10} = 0.468\)

Step 4: Write the final result

Thus, the number in standard notation is: 0.468 The given number in scientific notation, \(4.68 \times 10^{-1}\), is equal to 0.468 in standard notation.

Key Concepts

Scientific Notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is often used by scientists, engineers, and mathematicians to make calculations easier and to visually simplify complex numbers.

It consists of two parts: a coefficient and a power of ten. The coefficient must be a number between 1 and 10, and it is often a decimal. For instance, in the exercise's given number, the coefficient is 4.68. The second part is the exponent, which indicates the number of places to move the decimal point in the coefficient. A positive exponent moves the decimal point to the right, making the number larger, while a negative exponent, like in the provided example \(4.68 \times 10^{-1}\), moves it to the left, indicating a smaller number.

To convert a number from scientific notation to standard notation, you multiply the coefficient by ten raised to the power of the exponent. If the exponent is negative, you're essentially dividing by ten to the corresponding positive exponent.

Standard Notation

Standard notation is the regular way of writing numbers that we use daily. Unlike scientific notation, standard notation does not involve exponents; it simply refers to writing out the number fully, showing its value in terms of units, tens, hundreds, and so on.

Converting from scientific notation to standard notation involves moving the decimal point of the coefficient according to the power of ten. As in the exercise \(4.68 \times 10^{-1}\), the number is converted to 0.468 by moving the decimal one place to the left. Standard notation is especially handy for grasping the actual size or quantity represented by a number because it allows us to see all of the digits without the shorthand of scientific notation.

Exponents in Mathematics

Exponents, also referred to as powers, are a critical part of scientific notation and a foundational concept in mathematics. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, \(10^2\) is 10 multiplied by 10, which equals 100.

In the context of scientific notation, the base is always 10. A positive exponent shows that we have a large number, while a negative exponent, such as \(10^{-1}\), indicates a fractional or decimal number.A negative exponent, like the one in the exercise \(10^{-1}\), is equivalent to the reciprocal of the base raised to the opposite positive power. In this case, \(10^{-1}\) equals \(\frac{1}{10}\). Understanding the laws of exponents is crucial for working with scientific notation and for understanding the true scale of numbers expressed in this form.