Problem 60 - Algebra 1: Concepts and Skills

Question

Use a calculator to perform the indicated operation. Write the result in scientific notation and in decimal form. $$ (0.000094)^{3} $$

Step-by-Step Solution

Answer

The result of \((0.000094)^{3}\) in decimal form is 0.0000000000000088, and in scientific notation, it's \(8.8 \times 10^{-15}\).

Step 1: Calculate the Cubic Power

First, perform the calculation: \((0.000094)^{3}\). Depending on the type and setting of the calculator, the obtained result may already be in scientific notation.

Step 2: Write the Result in Decimal Form

After obtaining the result from step 1, if the result is in scientific notation, it needs to be written down in decimal form. This means expanding the notation out to show the actual decimal number.

Step 3: Convert the Decimal Form to Scientific Notation

Finally, convert the decimal form number to scientific notation. The purpose of scientific notation is to express very large or very small numbers in a simplified manner. It's in the form \(a \times 10^{b}\), where \(1 ≤ |a| < 10\) and \(b\) is an integer.

Key Concepts

Performing Operations Using Scientific Notation

Dealing with very large or very small numbers can be challenging, but scientific notation simplifies these tasks. When performing operations like multiplication or division with numbers in scientific notation, you mainly manipulate the coefficients (the number part) and the powers of ten separately. To \textbf{multiply} numbers in scientific notation, multiply the coefficients and add the exponents. For example, if you have to calculate \(2 \times 10^{3}) \cdot (5 \times 10^{2})\), you'd multiply 2 by 5 and add the exponents: \(10^{3+2}\), resulting in \(10^{5}\), finalizing it as \(1 \times 10^{5}\).

When \textbf{dividing}, you divide the coefficients and subtract the exponents of the powers of ten. For instance, for \(8 \times 10^{4}) \div (2 \times 10^{2})\), you divide 8 by 2 and subtract the exponent in the denominator from the exponent in the numerator: \(10^{4-2}\), which equals \(10^{2}\), and you would get \(4 \times 10^{2}\).\

To handle \textbf{raising a number in scientific notation to a power}, like \(0.000094)^{3}\) from our example, you raise both the coefficient and the power of ten to the given power. This can sometimes result in a coefficient that's not between 1 and 10, so additional adjustments are necessary to maintain proper scientific notation format.

Converting Decimal to Scientific Notation

Transforming a decimal into scientific notation requires two main steps. The first is to create a new number by moving the decimal point to the left or the right so that you have a number between 1 and 10; this number becomes the coefficient (\(a\)). The second step involves tracking the number of places you moved the decimal to determine the exponent (\(b\)). Each move to the left increases the exponent by 1, and each move to the right decreases the exponent by 1.

Let's take the number 0.000094 as an example. To convert this to scientific notation, we would move the decimal point five places to the right, which turns the number into 9.4 (which is between 1 and 10). To balance this change, we use a power of ten with an exponent equal to the negative number of places we moved the decimal, resulting in \(9.4 \times 10^{-5}\).

Conversely, if you have a large number such as 47000, you'd move the decimal point four places to the left to get 4.7, and your exponent would be positive 4, writing this number in scientific notation as \(4.7 \times 10^{4}\).

Calculating Powers of Ten

Calculating powers of ten is a fundamental skill when working with scientific notation, as it tells us how many places to move the decimal point. The general rule for \textbf{positive exponents} is that for \(10^{n}\), the decimal point is moved \(n\) places to the right. This denotes a number that is \(1\) followed by \(n\) zeros. For instance, \(10^{3}\) is 1,000.

When dealing with \textbf{negative exponents}, like \(10^{-n}\), the decimal point is shifted to the left, giving a value that is a fraction. For \(10^{-3}\), you'd write 0.001, which has three places to the right of the decimal point before the number 1 appears. This is because negative exponents indicate the division by that power of ten, thus making the number smaller.

Understanding powers of ten is crucial for accurately interpreting and adjusting numbers in scientific notation, ensuring they are in the correct form, with a coefficient between 1 and 10 and an appropriate power of ten. This skill allows students to convert back and forth between decimal and scientific notation with ease and precision.