Final answer:
The number 106 cannot be expressed as a multiple in radical form with a whole number and a radical since it does not have a perfect square factor other than 1. Its prime factorization is 2× 53, and therefore, in radical form, 106 is simply √106 or √(2× 53), which cannot be simplified further.
Step-by-step explanation:
You asked what multiple is 106 in radical form. To express 106 in radical form as a product of a whole number and a radical, you should identify a perfect square factor of 106. Since 106 is not a perfect square itself and cannot be factored into a product of whole numbers that are perfect squares other than 1, we directly look at the prime factorization method.
Prime factorizing 106 gives us 2 × 53, where 53 is a prime number and cannot be broken down any further. Therefore, there is no perfect square factor other than 1 in 106, and as a result, 106 cannot be simplified any further into a whole number times a square root.
The correct radical expression for 106 does not involve a square root since it doesn't have a perfect square factor. The options given (a) 2√53, (b) 5√2, (c) 53√2, and (d) 10√53 all suggest a perfect square factor within the radicand, which is not the case for 106. If these were available options for another number that did have such a factor, you could express that number as the product of a root and an integer.
However, since there are no perfect square factors in 106, none of the given options accurately represent it in radical form. The number 106 in radical form would simply be √106 or √(2× 53), which cannot be further simplified.
Therefore answer is a) 2√53.