The square-root symbol √ asks for the nonnegative number whose square is its operand. Thus √4 = 2. The cube-root symbol ∛ asks for the real number whose cube is its operand, so ∛8 = 2. The small three is an essential part of that second symbol.
Read what is already printed
In √(_) × _, the first square is already inside a root. Putting 2 there creates √2. Selecting the √2 expression key for that same square instead creates √(√2), a different value. Before choosing a root key, check whether the equation has supplied a root operator around that blank.
Equal roots can make a whole number
For a nonnegative a, √a × √a = a. In √(_) × _ + _ × 5 = 12, placing two in the last square contributes ten. The first product must be two. It can be √1 × 2, √4 × 1 or √2 × √2. The last option uses an irrational intermediate value but an exact whole-number product.
Cube roots and reciprocals
For real a and b, ∛a × ∛b = ∛(ab). Choosing a half and two as the radicands makes their product one, so ∛(1/2) × ∛2 = 1. That identity helps with ∛(_) × _ + _ × 8 = 33: a final four supplies thirty-two, leaving one for the root product.
A rule that does not work
Roots do not distribute across addition. √(a + b) is generally not √a + √b. For example, √(1 + 3) is two, while √1 + √3 is not two. Keep the entire printed operand under its root until you have evaluated it.
Check the domain
Mathoji uses real arithmetic. A negative value inside a square root is invalid. A cube root can be real for a negative operand, but the puzzle still restricts you to its displayed choices. Root expressions never permit division by zero.
When learning Master, simplify the familiar roots first, then look for pairs whose product cancels a root. Open a walkthrough only when you want a more specific nudge; the puzzle can be played without revealing its answers.