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Step-by-step explanation:

[tex]\huge{❥}{\mathtt{{\purple{\boxed{\tt{\pink{\red{A}\pink{n}\orange{s}\green{w}\blue{e}\purple{r᭄}}}}}}}}❥[/tex]

1) Given [tex]\sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{19}[/tex], we need to find [tex]x\sqrt{x} + \frac{1}{x\sqrt{x}}[/tex].

To find [tex]x\sqrt{x} + \frac{1}{x\sqrt{x}}[/tex], let's square the given equation.

[tex](\sqrt{x} + \frac{1}{\sqrt{x}})^2 = \sqrt{19}^2[/tex]

Expanding the left side and simplifying the right side, we get:

[tex]x + 2 + \frac{1}{x} = 19[/tex]

Rearranging the terms, we have:

[tex]x + \frac{1}{x} = 17[/tex]

To find [tex]x\sqrt{x} + \frac{1}{x\sqrt{x}}[/tex], let's square the equation we just found:

[tex](x+\frac{1}{x})^2 = 17^2[/tex]

Expanding and simplifying, we get:

[tex]x^2 + 2 + \frac{1}{x^2} = 289[/tex]

Now, let's rearrange the terms to express [tex]x\sqrt{x} + \frac{1}{x\sqrt{x}}[/tex]:

[tex]x^2 + \frac{1}{x^2} = 19[/tex]

Multiplying both sides by [tex]x[/tex], we get:

[tex]x^3 + 1 = 19x[/tex]

Now, subtracting [tex]\frac{1}{x^3}[/tex] from both sides, we get:

[tex]x^3 - \frac{1}{x^3} = 19x - \frac{1}{x^3}[/tex]

Thus, [tex]x^3 - \frac{1}{x^3} = 19x - \frac{1}{x^3}[/tex].

2) Given [tex]x - \frac{1}{x} = 11[/tex], we need to find [tex]x^3 - \frac{1}{x^3}[/tex].

To solve this, let's cube the given equation:

[tex](x - \frac{1}{x})^3 = 11^3[/tex]

Expanding the left side and simplifying the right side, we get:

[tex]x^3 - 3x + \frac{3}{x} - \frac{1}{x^3} = 1331[/tex]

Now, rearranging the terms, we have:

[tex]x^3 + \frac{1}{x^3} - 3(x - \frac{1}{x}) = 1331[/tex]

Given that [tex]x - \frac{1}{x} = 11[/tex], we can substitute this into the equation:

[tex]x^3 + \frac{1}{x^3} - 3(11) = 1331[/tex]

[tex]x^3 + \frac{1}{x^3} - 33 = 1331[/tex]

[tex]x^3 + \frac{1}{x^3} = 1364[/tex]

Therefore, [tex]x^3 - \frac{1}{x^3} = 1364[/tex].

EXPLANATION.

Question :

(1) (√x + 1/√x) = √19. - - - - - (1).

As we know that,

Formula of :

⇒ (a + b)² = a² + b² + 2ab.

Using this formula in this question, we get.

Squaring on both sides of the expression, we get.

⇒ (√x + 1/√x)² = (√19)².

⇒ (√x)² + (1/√x)² + 2(√x)(1/√x) = 19.

⇒ x + 1/x + 2 = 19.

⇒ x + 1/x = 19 - 2.

⇒ x + 1/x = 17. - - - - - (2).

To find : [x√x + 1/(x√x)].

Multiply equation (1) and equation (2), we get.

⇒ (√x + 1/√x)(x + 1/x) = 17√19.

⇒ √x(x + 1/x) + 1/√x(x + 1/x) = 17√19.

⇒ (x√x) + (√x/x) + (x/√x) + (1/x√x) = 17√19.

⇒ (x√x) + (1/√x) + (√x) + (1/x√x) = 17√19.

⇒ (x√x) + (1/x√x) + (√x + 1/√x) = 17√19.

Put the value of equation (1), we get.

⇒ (x√x) + (1/x√x) + (√19) = 17√19.

⇒ (x√x) + (1/x√x) = 17√19 - √19.

⇒ (x√x) + (1/x√x) = 16√19.

∴ The value of (x√x) + (1/x√x) = 16√19.

(2) (x - 1/x) = 11.

As we know that,

Formula of :

⇒ (a - b)³ = a³ - 3a²b + 3ab² - b³.

Using this formula in this question, we get.

Cubing on both sides of the expression, we get.

⇒ (x - 1/x)³ = (11)³.

⇒ (x)³ - 3(x²)(1/x) + 3(x)(1x²) - (1/x³) = 1331.

⇒ x³ - 3x + 3/x - 1/x³ = 1331.

⇒ x³ - 1/x³ - 3x + 3/x = 1331.

⇒ x³ - 1/x³ - 3(x - 1/x) = 1331.

Put the value of (x - 1/x) = 11 in the expression, we get.

⇒ x³ - 1/x³ - 3(11) = 1331.

⇒ x³ - 1/x³ - 33 = 1331.

⇒ x³ - 1/x³ = 1331 + 33.

⇒ x³ - 1/x³ = 1364.

∴ The value of (x³ - 1/x³) is equal to 1364.