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3. Given that [tex]\sqrt{2}=1.414[/tex], [tex]\sqrt{3}=1.732[/tex], [tex]\sqrt{5}=2.236[/tex], and [tex]\sqrt{10}=3.162[/tex], find the value to three decimal places of each of the following:

(i) [tex]\frac{2}{\sqrt{5}}[/tex]

(ii) [tex]\frac{2-\sqrt{3}}{\sqrt{3}}[/tex]

(iii) [tex]\frac{\sqrt{10}-\sqrt{5}}{\sqrt{2}}[/tex]

[tex]\[ [2010] \][/tex]

Sagot :

GinnyAnswer
Sure, let's solve each part step-by-step and find the required values to three decimal places.

### Part (i)

We need to find the value of [tex]\( \frac{2}{\sqrt{5}} \)[/tex].

Given that [tex]\( \sqrt{5} \approx 2.236 \)[/tex]:

[tex]\[ \frac{2}{\sqrt{5}} = \frac{2}{2.236} \][/tex]

Carrying out this division:

[tex]\[ \frac{2}{2.236} \approx 0.894 \][/tex]

So, the value of [tex]\( \frac{2}{\sqrt{5}} \)[/tex] to three decimal places is:

[tex]\[ 0.894 \][/tex]

### Part (ii)

We need to find the value of [tex]\( \frac{2 - \sqrt{3}}{\sqrt{3}} \)[/tex].

Given that [tex]\( \sqrt{3} \approx 1.732 \)[/tex]:

First, calculate [tex]\( 2 - \sqrt{3} \)[/tex]:

[tex]\[ 2 - \sqrt{3} = 2 - 1.732 = 0.268 \][/tex]

Then, we divide by [tex]\( \sqrt{3} \)[/tex]:

[tex]\[ \frac{0.268}{1.732} \approx 0.155 \][/tex]

So, the value of [tex]\( \frac{2 - \sqrt{3}}{\sqrt{3}} \)[/tex] to three decimal places is:

[tex]\[ 0.155 \][/tex]

### Part (iii)

We need to find the value of [tex]\( \frac{\sqrt{10} - \sqrt{5}}{\sqrt{2}} \)[/tex].

Given that [tex]\( \sqrt{10} \approx 3.162 \)[/tex], [tex]\( \sqrt{5} \approx 2.236 \)[/tex], and [tex]\( \sqrt{2} \approx 1.414 \)[/tex]:

First, calculate [tex]\( \sqrt{10} - \sqrt{5} \)[/tex]:

[tex]\[ \sqrt{10} - \sqrt{5} = 3.162 - 2.236 = 0.926 \][/tex]

Then, we divide by [tex]\( \sqrt{2} \)[/tex]:

[tex]\[ \frac{0.926}{1.414} \approx 0.655 \][/tex]

So, the value of [tex]\( \frac{\sqrt{10} - \sqrt{5}}{\sqrt{2}} \)[/tex] to three decimal places is:

[tex]\[ 0.655 \][/tex]

### Summary of the answers:

(i) [tex]\( \frac{2}{\sqrt{5}} \approx 0.894 \)[/tex]

(ii) [tex]\( \frac{2 - \sqrt{3}}{\sqrt{3}} \approx 0.155 \)[/tex]

(iii) [tex]\( \frac{\sqrt{10} - \sqrt{5}}{\sqrt{2}} \approx 0.655 \)[/tex]

These are the required values to three decimal places.
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