Question

Exercise (11) Given the numbers: x = 2√24+5√√54-9√6 and y = 4√75+√108-2√48-4√3. 1) Simplify x and y, then write x in the form a√6 and y in the form b√√3, where a and b are my natural numbers. (1 2) Compare x and y. Show your work process. 3) Let ABC be a triangle right angled at A such that: AB = x and AC = y. The unit of length is the centimeter. Calculate the exact value of the area of ABC, and write the answer in the form c√√2, where c is a natural number.​

Answer 1

To simplify the expressions x and y, we combine like terms using properties of square roots. After simplification, x is equal to 10√6 and y is equal to 12√3 + 6√2.

Step-by-step explanation:

To simplify the given expressions x and y, we need to combine like terms. Starting with x, we have:

x = 2√24 + 5√√54 - 9√6

Let's simplify the square roots:

x = 2√(4 * 6) + 5√(3 * 3 * 6) - 9√6

Using the property √(ab) = √a * √b, we can simplify further:

x = 2√4 * √6 + 5√3 * √3 * √6 - 9√6

We know that √4 = 2 and √3 * √3 = 3, so:

x = 2 * 2√6 + 5 * 3√6 - 9√6

Combining like terms, we get:

x = 4√6 + 15√6 - 9√6

x = 10√6

Now let's simplify y:

y = 4√75 + √108 - 2√48 - 4√3

Using the property √(ab) = √a * √b, we can simplify the square roots:

y = 4√(5 * 5 * 3) + √(2 * 2 * 3 * 3) - 2√(4 * 3) - 4√3

Simplifying further:

y = 4 * 5√3 + 2 * 3√2 - 2 * 2√3 - 4√3

Combining like terms, we get:

y = 20√3 + 6√2 - 4√3 - 4√3

y = 12√3 + 6√2

Therefore, x is simplified as 10√6 and y is simplified as 12√3 + 6√2.