Question

Given that (√3-5√2)(√3+2)=a+b√6. Find a and b. Show workings bit by bit

Answer 1

[AI-generated answer — Bot GPT 3.5]

First, we need to multiply (√3-5√2) by (√3+2). We can use the distributive property to do this:

(√3-5√2)(√3+2) = √3(√3+2) - 5√2(√3+2)

Next, we can simplify each term inside the brackets:

√3(√3+2) = (√3*√3) + (√3*2) = 3+2√3

-5√2(√3+2) = -5√2*√3 -5√2*2 = -5√6 -10√2

Now, we can combine the simplified terms:

(3+2√3) - (5√6 + 10√2)

Next, we can distribute the negative sign to each term inside the bracket:

3 + 2√3 - 5√6 - 10√2

Finally, we can rearrange the terms:

3 - 10√2 + 2√3 - 5√6

The final form is a+b√6, so we need to group like terms:

(3 - 10√2) + (2√3 - 5√6)

Therefore, a = 3 - 10√2 and b = 2√3 - 5√6.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the values of a and b in the expression (√3 - 5√2)(√3 + 2) = a + b√6, we can use the distributive property of multiplication.

Step 1: Expand the expression (√3 - 5√2)(√3 + 2).

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

Step 2: Simplify each term.

√3 * √3 = √(3 * 3) = √9 = 3
√3 * 2 = 2√3
-5√2 * √3 = -5√(2 * 3) = -5√6
-5√2 * 2 = -10√2

Step 3: Combine like terms.

3 + 2√3 - 5√6 - 10√2

Step 4: Rearrange the terms to collect the real number part and the irrational part.

(3 - 10√2) + (2√3 - 5√6)

So, a = 3 - 10√2 and b = 2√3 - 5√6.

Answer 3

[AI-generated answer — Explain Bot]

To find the values of a and b in the expression (√3 - 5√2)(√3 + 2) = a + b√6, we can multiply the two binomials using the distributive property.

Step 1: Expand the expression

Start by multiplying the terms in the first binomial with the terms in the second binomial:

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

Step 2: Simplify the multiplied terms

√3 * √3 simplifies to 3.
√3 * 2 simplifies to 2√3.
-5√2 * √3 simplifies to -5√6 because when multiplying radicals, you multiply the coefficients and combine the like radicals.
-5√2 * 2 simplifies to -10√2.

So, the expanded expression becomes:

(√3 - 5√2)(√3 + 2) = 3 + 2√3 - 5√6 - 10√2

Step 3: Combine like terms

Combine the terms with √3 and √6, as well as the numerical terms:

(√3 - 5√2)(√3 + 2) = (3 - 10√2) + (2√3 - 5√6)

Step 4: Simplify the result

Our goal is to write the expression in the form a + b√6. To do this, we need to regroup the terms.

Combine the numerical terms:
(3 - 10√2) = 3 - 10√2

Combine the terms with √6:
(2√3 - 5√6) = -5√6 + 2√3

So, the final expression is:

(√3 - 5√2)(√3 + 2) = 3 - 10√2 - 5√6 + 2√3

Comparing this to a + b√6, we can determine that:

a = 3 - 10√2
b = -5