Evaluate. Write your answer as a decimal or whole number.
$(3.1)^3=$
The goal is to evaluate the expression \((3.1)^3\), which represents the cube of the decimal \(3.1\). To solve this, we interpret the expression as the product of three factors of \(3.1\): \((3.1)^3 = 3.1 \times 3.1 \times 3.1\). We will break this calculation into two sequential multiplications.
First, we calculate the square of \(3.1\), which is \((3.1)^2 = 3.1 \times 3.1\).
Performing the multiplication:
\[3.1 \times 3.1 = 9.61\]
Next, we multiply this result by \(3.1\) once more to reach the final value:
\[(3.1)^2 \times 3.1 = 9.61 \times 3.1\]
We can compute this as:
\[9.61 \times 3.1 = 9.61 \times (3 + 0.1)\]
\[9.61 \times 3 = 28.83\]
\[9.61 \times 0.1 = 0.961\]
Adding these two results:
\[28.83 + 0.961 = 29.791\]
The calculation confirms that \((3.1)^3 = 29.791\). Thus, the final value is \(\boxed{29.791}\).
Understanding how to calculate powers of decimals is useful in many real-life applications, such as calculating compound interest or scaling physical measurements. For example, if you are planning to expand a storage cube where each side length is increased by a factor of \(3.1\), the new volume will be the cube of the original volume multiplied by \(3.1^3\). Mastering these operations ensures accuracy in financial projections, architectural designs, and scientific measurements.