Find the value of

$$\frac{\sqrt{1.24} imes \sqrt{2.79}}{\sqrt{2.64} imes \sqrt{5.94}}$$

Question

Find the value of

$$\frac{\sqrt{1.24} 	imes \sqrt{2.79}}{\sqrt{2.64} 	imes \sqrt{5.94}}$$

Accepted Answer

**Solution:**

1. Start with the given expression:
   $$\frac{\sqrt{1.24} 	imes \sqrt{2.79}}{\sqrt{2.64} 	imes \sqrt{5.94}}$$

2. Combine the square roots in numerator and denominator:
   $$= \frac{\sqrt{1.24 	imes 2.79}}{\sqrt{2.64 	imes 5.94}} = \sqrt{\frac{1.24 	imes 2.79}{2.64 	imes 5.94}}$$

3. Convert decimals to fractions or multiply directly:
   - Multiply numerator: $1.24 	imes 2.79 = 3.4596$
   - Multiply denominator: $2.64 	imes 5.94 = 15.6816$

4. So expression becomes:
   $$\sqrt{\frac{3.4596}{15.6816}}$$

5. Simplify the fraction inside the root:
   $$\frac{3.4596}{15.6816} = \frac{34596}{156816}$$ (multiplying numerator and denominator by 10000)

6. Simplify the fraction by dividing numerator and denominator by 1116:
   $$\frac{34596 \div 1116}{156816 \div 1116} = \frac{31}{140.5}$$

But 140.5 is not a whole number, so let's try another approach.

**Alternative approach using factorization:**

- Express decimals as fractions:
  - $1.24 = \frac{124}{100} = \frac{31}{25}$
  - $2.79 = \frac{279}{100}$
  - $2.64 = \frac{264}{100} = \frac{66}{25}$
  - $5.94 = \frac{594}{100} = \frac{297}{50}$

- Now rewrite the expression:
  $$\frac{\sqrt{\frac{31}{25}} 	imes \sqrt{\frac{279}{100}}}{\sqrt{\frac{66}{25}} 	imes \sqrt{\frac{297}{50}}} = \frac{\sqrt{\frac{31 	imes 279}{25 	imes 100}}}{\sqrt{\frac{66 	imes 297}{25 	imes 50}}}$$

- Simplify inside the roots:
  $$= \sqrt{\frac{31 	imes 279}{25 	imes 100} 	imes \frac{25 	imes 50}{66 	imes 297}} = \sqrt{\frac{31 	imes 279 	imes 25 	imes 50}{25 	imes 100 	imes 66 	imes 297}}$$

- Cancel $25$:
  $$= \sqrt{\frac{31 	imes 279 	imes 50}{100 	imes 66 	imes 297}}$$

- Simplify $\frac{50}{100} = \frac{1}{2}$:
  $$= \sqrt{\frac{31 	imes 279}{2 	imes 66 	imes 297}}$$

- Factor numerator and denominator:
  - $279 = 3 	imes 3 	imes 31$ (since $279 = 3^2 	imes 31$)
  - $297 = 3 	imes 3 	imes 33$ (since $297 = 3^3 	imes 11$ actually $297 = 3^3 	imes 11$)

- Substitute:
  $$= \sqrt{\frac{31 	imes 3^2 	imes 31}{2 	imes 66 	imes 3^3 	imes 11}}$$

- Simplify numerator:
  $$31 	imes 31 = 31^2$$

- Simplify denominator:
  $$66 = 6 	imes 11 = 2 	imes 3 	imes 11$$

- So denominator:
  $$2 	imes 66 	imes 3^3 	imes 11 = 2 	imes (2 	imes 3 	imes 11) 	imes 3^3 	imes 11 = 2 	imes 2 	imes 3 	imes 11 	imes 3^3 	imes 11 = 4 	imes 3^4 	imes 11^2$$

- Numerator:
  $$31^2 	imes 3^2$$

- So expression inside root:
  $$= \sqrt{\frac{31^2 	imes 3^2}{4 	imes 3^4 	imes 11^2}} = \sqrt{\frac{31^2}{4 	imes 3^{2} 	imes 11^2}}$$

- Simplify powers of 3:
  $$3^2 	ext{ in numerator and } 3^4 	ext{ in denominator leaves } 3^2 	ext{ in denominator}$$

- So:
  $$= \sqrt{\frac{31^2}{4 	imes 3^{2} 	imes 11^2}} = \frac{31}{2 	imes 3 	imes 11} = \frac{31}{66}$$

**Answer:** $\frac{31}{66}$

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