Question

Rationalies the denominator and simplify:4√3+5√2
/√48+√18​

Answer 1

Answer:

(9+4√6)/15

Step-by-step explanation:

(4√3+5√2)/(√48+√18)

=(4√3+5√2)/(√48+√18)×(√48-√18)/(√48-√18)

={(4√3×√48)+(5√2×√48)-(4√3×√18)-(5√2×√18)}/{(√48)^2-(√18)^2}

=4√3×4√3+5√2×4√3-4√3×3√2-5√2×3√2/48-18

=16×3+20√6-12√6-15×2/30

={48+20√6-12√6-30}/30

=18+8√6/30

=9+4√6/15

Answer 2

Given:

The expression:-

• $\bf \dfrac{4\sqrt{3} + 5\sqrt{2}}{\sqrt{48} +\sqrt{18}}$

What To Find:

We have to -

• Rationalise the denominator and simplify the expression.

Solution:

• Finding the rationalising factor.

Here, the rationalising factor of the denominator is -

$\sf \to \: \sqrt{48} - \sqrt{18}$

• Multiplying with the expression.

$\sf \to \dfrac{4\sqrt{3} + 5\sqrt{2}}{\sqrt{48} +\sqrt{18}} \times \dfrac{\sqrt{48} - \sqrt{18}}{\sqrt{48} - \sqrt{18}}$

Take them as common,

$\sf \to \dfrac{(4\sqrt{3} + 5\sqrt{2}) \times (\sqrt{48} - \sqrt{18}) }{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

• Solving the denominator.

$\sf \to \dfrac{(4\sqrt{3} + 5\sqrt{2}) \times (\sqrt{48} - \sqrt{18}) }{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

Written them as,

$\sf \to \dfrac{4\sqrt{3}(\sqrt{48} - \sqrt{18}) + 5\sqrt{2}(\sqrt{48} - \sqrt{18})}{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

Solve the first brackets,

$\sf \to \dfrac{48 - 12\sqrt{6} + 5\sqrt{2}(\sqrt{48} - \sqrt{18})}{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

Solve the second brackets,

$\sf \to \dfrac{48 - 12\sqrt{6} - 20\sqrt{6} - 30}{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

Solve the terms,

$\sf \to \dfrac{18 - 8\sqrt{6}}{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

• Solving the denominator.

$\sf \to \dfrac{18 - 8\sqrt{6}}{(\sqrt{48} +\sqrt{18}) \times (\sqrt{48} - \sqrt{18})}$

Use the identity (a - b) (a + b) = a² - b²

$\sf \to \dfrac{18 - 8\sqrt{6}}{(\sqrt{48})^2 + (\sqrt{18})^2}$

Find the squares,

$\sf \to \dfrac{18 - 8\sqrt{6}}{48-18}$

Subtract the values,

$\sf \to \dfrac{18 - 8\sqrt{6}}{30}$

• Simplifying the expression.

$\sf \to \dfrac{18 - 8\sqrt{6}}{30}$

Here we can see that 2 is common in the numerator so let's take it out,

$\sf \to \dfrac{2(9 - 4\sqrt{6})}{30}$

Cancel 2 and 30,

$\sf \to \dfrac{9 - 4\sqrt{6}}{15}$

Final Answer:

∴ Thus, the answer is:-

• $\bf \dfrac{9 - 4\sqrt{6}}{15}$