Algebra

sqrt(7) * sqrt(35)

Step-by-step solution with explanation

Final Answer

Step-by-step solution

1

Write the product under one radical

When multiplying two square roots, we can combine them under a single square root. This is the product rule for radicals.
2

Multiply the numbers inside

Simply multiply 7 times 35 to get 245. Now we need to simplify this square root.
3

Factor 245 to find perfect squares

We look for a perfect square factor of 245. Since 49 = 7², it divides evenly into 245, giving us 49 × 5.
4

Split the radical and simplify

We break the radical into two parts. The square root of 49 is exactly 7, so it comes out of the radical. The 5 stays inside since it has no perfect square factors.

Understanding this problem

Learning Insight

The product rule for radicals works because square roots and squaring are inverse operations. When you write √a × √b = √(ab), you are using the fact that (√a × √b)² = a × b, so √a × √b must equal √(ab). Finding perfect square factors is key to simplifying — always look for the largest perfect square that divides your number.

Quick Tip

To quickly find perfect square factors, try dividing by 4, 9, 16, 25, 36, 49, etc. in order. The moment one divides evenly, you have your factor and can simplify.

Common Mistake

A common mistake is stopping at √245 without simplifying further. Students often miss that 49 (a perfect square) is a factor of 245, leaving the answer in an unsimplified form.