. Simplify the following expressions points)

a. sqrt(20) + sqrt(5)
b. (6 + sqrt(18))/6
c. sqrt(9y^4x^0.5)^0.5

a. To simplify the expression sqrt(20) + sqrt(5), we can simplify each square root separately and then add the results.

First, let's simplify sqrt(20). We can rewrite 20 as the product of its largest perfect square factor and the remaining factor:

20 = 4 * 5

sqrt(20) = sqrt(4 * 5)

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

sqrt(20) = sqrt(4) * sqrt(5)

Since sqrt(4) = 2, we have:

sqrt(20) = 2 * sqrt(5)

Now, let's simplify sqrt(5). Since there are no perfect square factors in 5, we cannot simplify it further.

Therefore, the simplified expression is:

sqrt(20) + sqrt(5) = 2 * sqrt(5) + sqrt(5)

To combine like terms, we can rewrite sqrt(5) as 1 * sqrt(5):

2 * sqrt(5) + sqrt(5) = 2 * sqrt(5) + 1 * sqrt(5)

Now we can add the like terms:

2 * sqrt(5) + 1 * sqrt(5) = (2 + 1) * sqrt(5) = 3 * sqrt(5)

So the simplified expression is 3 * sqrt(5).

b. To simplify the expression (6 + sqrt(18))/6, we can simplify the square root and then divide.

First, let's simplify sqrt(18). We can rewrite 18 as the product of its largest perfect square factor and the remaining factor:

18 = 9 * 2

sqrt(18) = sqrt(9 * 2)

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

sqrt(18) = sqrt(9) * sqrt(2)

Since sqrt(9) = 3, we have:

sqrt(18) = 3 * sqrt(2)

Now, let's rewrite the expression:

(6 + sqrt(18))/6 = 6/6 + sqrt(18)/6

Simplifying the fraction, we have:

6/6 + sqrt(18)/6 = 1 + (3 * sqrt(2))/6

Simplifying further, we can divide the numerator and denominator of (3 * sqrt(2))/6 by 3:

1 + (3 * sqrt(2))/6 = 1 + (sqrt(2))/2

So the simplified expression is 1 + (sqrt(2))/2.

c. To simplify the expression sqrt(9y^4x^0.5)^0.5, we can simplify the square root and the exponent.

First, let's simplify sqrt(9y^4x^0.5). We can rewrite it as:

sqrt(9y^4x^0.5) = sqrt(9) * sqrt(y^4) * sqrt(x^0.5)

Since sqrt(9) = 3 and sqrt(y^4) = y^2, we have:

sqrt(9y^4x^0.5) = 3 * y^2 * x^0.25

Now, let's simplify the exponent. We can rewrite x^0.25 as the square root of x:

x^0.25 = sqrt(x)

Now, let's rewrite the expression:

sqrt(9y^4x^0.5)^0.5 = (3 * y^2 * sqrt(x))^0.5

To simplify the exponent, we can take the square root of each term:

(3 * y^2 * sqrt(x))^0.5 = 3^0.5 * (y^2)^0.5 * (sqrt(x))^0.5

Since 3^0.5 = sqrt(3) and (y^2)^0.5 = y, we have:

3^0.5 * (y^2)^0.5 * (sqrt(x))^0.5 = sqrt(3) * y * sqrt(sqrt(x))

So the simplified expression is sqrt(3) * y * sqrt(sqrt(x)).

a. To simplify the expression sqrt(20) + sqrt(5), we can first simplify the individual square roots. The square root of 20 can be written as the square root of 4 times 5. Since the square root of 4 is 2, we can rewrite sqrt(20) as 2 * sqrt(5).

So now our expression becomes 2 * sqrt(5) + sqrt(5). Since both terms have the same radical, sqrt(5), we can combine them by adding their coefficients. The coefficient of the first term is 2, and the coefficient of the second term is 1. Adding them together gives us 2 + 1 = 3. Therefore, the simplified expression is 3 * sqrt(5).

b. To simplify the expression (6 + sqrt(18))/6, we can start by simplifying the square root. The square root of 18 can be expressed as the square root of 9 times 2. The square root of 9 is 3, so we can rewrite sqrt(18) as 3 * sqrt(2).

Now our expression becomes (6 + 3 * sqrt(2))/6. We can further simplify this expression by dividing both terms by 3. Dividing 6 by 3 gives us 2, and dividing 3 * sqrt(2) by 3 gives us sqrt(2).

Therefore, the simplified expression is 2 + sqrt(2).

c. The expression sqrt(9y^4x^0.5)^0.5 can be simplified by applying the exponent rule for radicals. According to this rule, when we raise a square root to a power, we can simplify by removing the square root sign and raising the radicand to the power.

In this case, the exponent outside the square root is 0.5. So we can simplify the expression by raising the radicand inside the square root, 9y^4x^0.5, to the power of 0.5.

Raising 9 to the power of 0.5 gives us sqrt(9) = 3.
Raising y^4 to the power of 0.5 gives us (y^4)^0.5 = y^2.
Raising x^0.5 to the power of 0.5 gives us (x^0.5)^0.5 = x^0.25.

Therefore, the simplified expression is 3y^2x^0.25.

1) change sqrt(20) to sqrt(4) sqrt(5)

2) change sqrt(18) to sqrt(9) sqrt 2