Given:
x = 7 √40

To find:
The value of √x and 1/√x

Solution:

Step 1: Simplifying x
We are given that x = 7 √40. Let's simplify this expression.

To simplify √40, we need to find the largest square number that divides 40.
The largest square number that divides 40 is 4, as 4 is the square of 2.

Now, we can rewrite √40 as √(4 * 10).

Using the property of square roots, we can split the square root as the product of the square roots of the individual factors:

√(4 * 10) = √4 * √10

Simplifying further:
√4 = 2 (since 2 * 2 = 4)
√10 remains as it is.

Therefore, x = 7 * 2 * √10 = 14 √10.

Step 2: Finding √x
To find the value of √x, we need to take the square root of x.

√x = √(14 √10)

Using the property of square roots, we can split the square root as the product of the square roots of the individual factors:

√(14 √10) = √14 * √(√10)

Simplifying further:
√14 remains as it is.

For √(√10), we can rewrite it as (√10)^(1/2).

Using the property of exponents, we multiply the exponents:
(√10)^(1/2) = 10^(1/4)

Therefore, √x = √14 * 10^(1/4).

Step 3: Finding 1/√x
To find the value of 1/√x, we need to take the reciprocal of √x.

1/√x = 1/(√14 * √(√10))

Using the property of reciprocals, we can split the fraction as the product of the reciprocals of the individual factors:

1/(√14 * √(√10)) = 1/√14 * 1/√(√10)

Simplifying further:
1/√14 remains as it is.

For 1/√(√10), we can rewrite it as (√10)^(-1/2).

Using the property of exponents, we change the sign of the exponent:
(√10)^(-1/2) = 10^(-1/4)

Therefore, 1/√x = 1/√14 * 10^(-1/4).

Final Answer:
- √x = √14 * 10^(1/4)
- 1/√x = 1/√14 * 10^(-1/4)