Math, asked by raghu6892, 10 months ago

under root 2 minus under root 3 / 3 under root 2 + 2 under root 3 is equal to a + b under root 6 then find the value of a and b​

Answers

Answered by crfa981
0

o square a number, just multiply it by itself ...

Example: What is 3 squared?

3 Squared = 3x3 box = 3 × 3 = 9

"Squared" is often written as a little 2 like this:

4 Squared equals 16

This says "4 Squared equals 16"

(the little 2 means the number appears twice in multiplying, so 4×4=16)

Square Root

A square root goes the other direction:

square root of 9 is 3

3 squared is 9, so a square root of 9 is 3

It is like asking:

What can I multiply by itself to get this?

Definition

Here is the definition:

A square root of x is a number r whose square is x:

r2 = x

r is a square root of x

The Square Root Symbol

radical symbol    

This is the special symbol that means "square root", it is like a tick,

and actually started hundreds of years ago as a dot with a flick upwards.

It is called the radical, and always makes mathematics look important!

We can use it like this:

square root of 9

we say "square root of 9 equals 3"

Example: What is √36 ?

Answer: 6 × 6 = 36, so √36 = 6

Negative Numbers

We can also square negative numbers.

Example: What is minus 5 squared?

But hang on ... what does "minus 5 squared" mean?  

square the 5, then do the minus?

or square (−5)?

It isn't clear! And we get different answers:

square the 5, then do the minus: −(5×5) = −25

square (−5): (−5)×(−5) = +25

So let's make it clear by using "( )".

Example Corrected: What is (−5)2 ?

Answer:

(−5) × (−5) = 25

(because a negative times a negative gives a positive)

That was interesting!

When we square a negative number we get a positive result.

Just the same as when we square a positive number:

5x5 = -5x-5

Now remember our defintion of a square root?

A square root of x is a number r whose square is x:

r2 = x

r is a square root of x

And we just found that:

(+5)2 = 25

(−5)2 = 25

So both +5 and −5 are square roots of 25

Two Square Roots

There can be a positive and negative square root!

This is important to remember.

Example: Solve w2 = a

Answer:

w = √a   and   w = −√a

Principal Square Root

So if there are really two square roots, why do people say √25 = 5 ?

Because √ means the principal square root ... the one that isn't negative!

There are two square roots, but the symbol √ means just the principal square root.

Example:

The square roots of 36 are 6 and −6

But √36 = 6 (not −6)

The Principal Square Root is sometimes called the Positive Square Root (but it can be zero).

Plus-Minus Sign

±  is a special symbol that means "plus or minus",

   

so instead of writing:   w = √a   and   w = −√a

we can write:   w = ±√a

In a Nutshell

When we have: r2 = x

then: r = ±√x

Why Is This Important?

Why is this "plus or minus" important? Because we don't want to miss a solution!

Example: Solve x2 − 9 = 0

Start with: x2 − 9 = 0

Move 9 to right: x2 = 9

Square Roots: x = ±√9

Answer: x = ±3

The "±" tells us to include the "−3" answer also.

x^2-9

Example: Solve for x in (x − 3)2 = 16

Start with: (x − 3)2 = 16

Square Roots: x − 3 = ±√16

Calculate √16: x − 3 = ±4

Add 3 to both sides: x = 3 ± 4

Answer: x = 7 or −1

Check: (7−3)2 = 42 = 16

Check: (−1−3)2 = (−4)2 = 16

Square Root of xy

When two numbers are multiplied within a square root, we can split it into a multiplication of two square roots like this:

√xy = √x√y

but only when x and y are both greater than or equal to 0

 

Example: What is √(100×4) ?

√(100×4) = √(100) × √(4)

 = 10 × 2

 = 20

And √x√y = √xy :

Example: What is √8√2 ?

√8√2 = √(8×2)

 = √16

 = 4

Example: What is √(−8 × −2) ?

√(−8 × −2) = √(−8) × √(−2)

 = ???

We seem to have fallen into some trap here!

We can use Imaginary Numbers, but that leads to a wrong answer of −4

Oh that's right ...

The rule only works when x and y are both greater than or equal to 0

So we can't use that rule here.

Instead just do it this way:

√(−8 × −2) = √16 = +4

Why does √xy = √x√y ?

We can use the fact that squaring a square root gives us the original value back again:

(√a)2 = a

Assuming a is not negative!

We can do that for xy: (√xy)2 = xy

And also to x, and y, separately: (√xy)2 = (√x)2(√y)2

Use a2b2 = (ab)2: (√xy)2 = (√x√y)2

Remove square from both sides: √xy = √x√y

An Exponent of a Half

A square root can also be written as a fractional exponent of one-half:

square-root-exponent-half

but only for x greater than or equal to 0

How About the Square Root of Negatives?

The result is an Imaginary Number... read that page to learn more.

 

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