how do i simpilify a square root? example: 4*(sqrt54)?

Question

please explain, and is it 12(sqrt6)?

Answer 1

You are correct. And, here’s why.

First, the “square root” of a number is simply that number raised to the ½ power. For example: 4^½ = 2

If you think about it,it makes sense. There's a rule of exponents that (m^p)^q = m^(pq), so
(4^½)² = 4^[(½)(2)] = 4^1 = 4, If you "square" the "square root" of a number you get the number. Right? Unfortunately it doesn't work so well the other way. That is, (4²)^½) gives you not just 4, butt ±4.

Anyway, because the"square root" is really an exponent (½), it obeys the rules of exponents. The particular rule we’re interested in here is (mn)^p = (m^p)(n^p).

54^½ = (9^½)(6^½)
Since 9^½=3
54^½ = (9^½)(6^½) = (3)(6^½)

Since the problem is (4)(54^½)
(4)(54^½) = (4)( 3)(6^½) = (12)(6^½) or 12√6

Answer 2

To simplify the square root, you want to factor the number under the square root sign. If you have any pairs of factors that are the same, then you have a square, and they can be extracted from under the square root sign.

In your example, 54 is the number under the square root sign. It can be factored as 2 * 3 * 3 * 3. You have one pair of 3's (and one extra 3), so you can extract a 3 from under the square root sign, leaving you with 6 under the square root sign, and 4 * 3 outside.

Another way of looking at it is:

4 * sqrt(54) = 4 * sqrt(6 * 9) = 4 * sqrt(9) * sqrt(6) = 4 * 3 * sqrt(6) = 12 * sqrt(6)

So your answer is correct.

Hope this helps.

Answer 3

4√54 = 12√6 is correctly simplified, you just look for square numbers inside the root sign

Ex

√180

= √9√20

= √9√4√5

= 3.2.√5

= 6√5

Answer 4

= 4 x √(9 x 6)
= 4 x 3√6
= 12√6 (as you suggest)

Answer 5

factorise 54 = 2* 27 = 2*3^3
for taking square root take one from each pair and leave the unpaired one

2^1- 2^0 taken left 2
3^3 3 (from 3^2) left 3

multiply the things taken 2^0*3^1 = 3

multiply the thing left out 6

so sqrt(54) = 3 sqrt(6)
multiply by 4 to get 4sqrt(54) = 12sqrt(6)

Answer 6

: 4*(sqrt54)= 4*(sqrt9*6)=4*3: (sqrt6)answer.=12: (sqrt6) yes