Problem 53 - Algebra 1: Concepts and Skills

Question

Perform the indicated operation without using a calculator. Write the result in scientific notation. $$ \left(3 \times 10^{2}\right)^{3} $$

Step-by-Step Solution

Answer

The result of the operation \(\left(3 \times 10^{2}\right)^{3}\) written in scientific notation is \(2.7 \times 10^{7}\)

Step 1: Cube the base and the exponent separately

Cube the base number, 3, and also cube the power of ten. This is possible due to the exponentiation rule \((a^m)^n = a^{mn}\).\[ (3 \times 10^{2})^{3} = 3^3 \times (10^2)^3 \]

Step 2: Perform the cubing

Cubing 3 gives us 27 and cubing \(10^{2}\) gives us \(10^{6}\) because \(3^3=27\) and \((10^{2})^{3} = 10^6\]

Step 3: Write the result in scientific notation

Scientific notation requires that the number before the power of ten be between one and ten. Therefore, the result of 27 needs to be changed to 2.7 and the exponent adjusted accordingly. Hence, 27 can be rewritten as \(2.7 \times 10^{1}\) which when multiplied by \(10^{6}\) gives \(2.7 \times 10^{7}\)

Key Concepts

Exponentiation Rule

Understanding the exponentiation rule is essential in algebra to simplify expressions involving powers. It states that when raising one power to another, we multiply the exponents. For example, when we have an expression like \( (a^m)^n \), we can simplify it to \( a^{mn} \). This rule makes calculating powers much easier and more efficient.

In the context of our exercise \( (3 \times 10^{2})^{3} \), we can see the exponentiation rule in action. Here, both the base number 3 and the exponent \( 10^{2} \) are raised to the power of 3. This gives us \( 3^3 \times (10^2)^3 \), which simplifies to \( 3^9 \times 10^{6} \). The exponentiation rule helps us avoid lengthy calculations that would otherwise be needed if we multiplied the base number and the power of ten separately.

Cubing Numbers

Cubing a number is a specific case of exponentiation where the exponent is 3, essentially raising a number to the third power. When we cube a number, we are calculating the volume of a cube with sides of equal length. For instance, \( 3^3 \) represents a cube with a side of length 3 units and the volume would be 27 cubic units.

The key to cubing is to understand that it means multiplying the number by itself twice more (three times in total). So, in our exercise, to cube the number 3, we calculate \( 3 \times 3 \times 3 \) which equals 27. This operation gives us the volume or the total number of units within a cube, and it also gives us a more profound comprehension of how numbers grow exponentially. Finding cube roots is also an essential skill, as it involves discovering the number which, when cubed, results in the given number.

Expressing in Scientific Notation

Scientific notation is a way to express very large or very small numbers conveniently. It consists of two parts: a number between 1 and 10, and a power of ten. This method is particularly useful in fields like science and engineering, where working with extreme values is common. To write a number in scientific notation, you will typically move the decimal place until you have a number between 1 and 10 and count the places you moved the decimal to determine the exponent used with the power of ten.

In our example, we needed to express the result \(27 \times 10^{6}\) in scientific notation. Since 27 is not between 1 and 10, we convert it to \(2.7 \times 10^{1}\). Now, \(2.7\) is within our desired range. We then combine it with \(10^{6}\) to get the final answer in scientific notation, \(2.7 \times 10^{7}\). This format is not only more concise but also easily digestible especially when dealing with very large numbers.