Final answer:
The expression 6√20 - 3√45 simplifies to 12√5 - 9√5, which then simplifies further to 3√5. This is achieved by first factoring the numbers inside the square roots to find and take out the square factors.
Step-by-step explanation:
The question is asking to simplify the expression 6√20 - 3√45. To simplify this expression, we first need to simplify the square roots by factoring them into prime factors and identifying perfect square factors within them. The number 20 can be written as 4×5, which means √20 = √(4×5) = √4 × √5 = 2√5. Similarly, the number 45 can be factored into 9×5, and thus √45 = √(9×5) = √9 × √5 = 3√5.
To simplify the expression completely, we can start by simplifying each square root individually.
For √20, we can break it down into √4 * √5. Since √4 is 2, we can rewrite √20 as 2√5.
For √45, we can break it down into √9 * √5. Since √9 is 3, we can rewrite √45 as 3√5.
Therefore, the expression simplifies to 6(2√5) - 3(3√5).
Simplifying further, we get 12√5 - 9√5. Finally, combining like terms, we have 3√5.
Now substituting these simplified square roots back into the initial expression, we get:
6×2√5 - 3×3√5
12√5 - 9√5
The square roots of 5 are like terms, so we can combine them by subtraction:
12√5 - 9√5 = 3√5.
This is the simplified form of the initial expression.