Question: Finding nth roots of perfect nth powers with signs Evaluate the following. Click on “Not a real…

Finding nth roots of perfect nth powers with signs Evaluate the following. Click on “Not a real number” if applicable. (a) -\sqrt[4]{625} = \boxed{\quad} (b) \sqrt[3]{-64} = \boxed{\quad} Not a real number

Solution

Evaluate the following expressions: (a) \(-\sqrt[4]{625}\) First, find the fourth root of 625. \[ \sqrt[4]{625} = \sqrt[4]{5^4} = 5 \] Apply the negative sign: \[ -\sqrt[4]{625} = -5 \] So the result for (a) is \(-5\). (b) \(\sqrt[3]{-64}\) Since the cube root of a negative number is real, find the cube root of \(-64\). \[ \sqrt[3]{-64} = \sqrt[3]{-(4^3)} = -4 \] So the result for (b) is \(-4\).