1. The problem asks to find the geometric mean of 4 and 21. 2. The formula for the geometric mean of two numbers $a$ and $b$ is: $$\text{Geometric Mean} = \sqrt{a \times b}$$ 3. Substitute $a=4$ and $b=21$ into the formula: $$\sqrt{4 \times 21}$$ 4. Multiply inside the square root: $$\sqrt{84}$$ 5. Simplify $\sqrt{84}$ by factoring 84 into its prime factors: $$84 = 4 \times 21 = 4 \times 3 \times 7$$ 6. Since $4$ is a perfect square, we can take its square root out: $$\sqrt{84} = \sqrt{4 \times 21} = \sqrt{4} \times \sqrt{21} = 2 \sqrt{21}$$ 7. Therefore, the geometric mean of 4 and 21 is: $$2 \sqrt{21}$$