Question

step - by - step solution
step 1: what is the index on the radical?
the index on the radical is 2.
because the index is 2, write each factor of the radicand as the product of two factors, one of which is a perfect square.
the perfect squares are 1, 4, 9, 16, 25, ...
the largest perfect square factor of 112 is \square.

Explanation:

Step1: Factorize 112

First, we factorize 112. We know that \(112 = 16\times7\), and also \(112 = 4\times28\), \(112 = 2\times56\), etc.

Step2: Identify perfect square factors

Now, we check which of these factors are perfect squares. The perfect squares we consider are \(1, 4, 9, 16, 25, \dots\). Among the factors of 112, 16 is a perfect square (\(4^2 = 16\)) and 4 is also a perfect square (\(2^2=4\)). Since we need the largest perfect square factor, we compare 16 and 4. Clearly, 16 is larger than 4. And we can see that \(112 = 16\times7\), where 16 is a perfect square and 7 is not a perfect square. So the largest perfect square factor of 112 is 16.

Answer:

16