What is $\sqrt[3]{-64}$?
The objective is to calculate the value of the expression \(\sqrt[3]{-64}\). The cube root of a real number \(x\), denoted \(\sqrt[3]{x}\), is the unique real number \(y\) such that \(y^3 = x\). Since the cube function \(f(y) = y^3\) is strictly increasing, there is exactly one real cube root for every real number, including negative ones. In this case, we are looking for a real number \(y\) such that \(y^3 = -64\).
We use the property that for any positive real number \(k\), the cube root of its negative, \(-k\), is equal to the negative of the cube root of \(k\). That is, \(\sqrt[3]{-k} = -\sqrt[3]{k}\). Here, we have \(k = 64\), so the expression becomes \(\sqrt[3]{-64} = -\sqrt[3]{64}\). We then search for a number whose cube is \(64\). Since \(4 \times 4 \times 4 = 16 \times 4 = 64\), we find that \(\sqrt[3]{64} = 4\).
Substituting the result back into our equation, we have \(\sqrt[3]{-64} = -(4) = -4\). To verify, we can check that \((-4)^3 = (-4) \times (-4) \times (-4) = 16 \times (-4) = -64\). This confirms that the value is indeed \(-4\).
Cube roots are frequently used in geometry and engineering. For instance, if a manufacturing company needs to create a storage container in the shape of a cube with a specific volume, they must calculate the cube root of the volume to determine the required side length. In a theoretical context involving negative values, finding cube roots helps in understanding transformations in coordinate geometry, where values might be flipped across the axes. Knowing that \(\sqrt[3]{-64} = -4\) illustrates the property that negative volumes or dimensions can be represented algebraically to maintain consistency in complex mathematical models.