Understanding the Mathematical Problem

The problem asks us to evaluate a complex expression involving square roots and fractions, calculate its value (let's call it \(x\)), and then find the square root of that value, i.e., \(\sqrt{x}\). The expression is given as:

\(x = \left( {\frac{{√ {1875} }}{{√ {3888} }} \div \frac{{√ {1200} }}{{\sqrt 768}}} \right) \times \frac{{√ {175} }}{{√ {1792} }}\)

To solve this, we need to simplify each square root term first by factoring out perfect squares. Then, we will perform the division and multiplication operations in the correct order.

Simplifying the Square Roots

Let's simplify each square root term involved in the expression for \(x\):

• \(\sqrt{1875}\): \(1875 = 25 \times 75 = 25 \times 25 \times 3 = 625 \times 3\). So, \(\sqrt{1875} = \sqrt{625 \times 3} = \sqrt{625} \times \sqrt{3} = 25\sqrt{3}\).
• \(\sqrt{3888}\): \(3888 = 2 \times 1944 = 2 \times 2 \times 972 = 4 \times 2 \times 486 = 8 \times 2 \times 243 = 16 \times 243\). \(243 = 3 \times 81 = 3 \times 9^2\). So, \(3888 = 16 \times 3 \times 81 = 16 \times 3 \times 9^2 = (4^2) \times 3 \times (9^2) = (4 \times 9)^2 \times 3 = 36^2 \times 3\). So, \(\sqrt{3888} = \sqrt{36^2 \times 3} = \sqrt{36^2} \times \sqrt{3} = 36\sqrt{3}\).
• \(\sqrt{1200}\): \(1200 = 12 \times 100 = 4 \times 3 \times 100 = 4 \times 100 \times 3 = 400 \times 3\). So, \(\sqrt{1200} = \sqrt{400 \times 3} = \sqrt{400} \times \sqrt{3} = 20\sqrt{3}\).
• \(\sqrt{768}\): \(768 = 3 \times 256 = 3 \times 16^2\). So, \(\sqrt{768} = \sqrt{3 \times 16^2} = \sqrt{16^2} \times \sqrt{3} = 16\sqrt{3}\).
• \(\sqrt{175}\): \(175 = 25 \times 7 = 5^2 \times 7\). So, \(\sqrt{175} = \sqrt{5^2 \times 7} = \sqrt{5^2} \times \sqrt{7} = 5\sqrt{7}\).
• \(\sqrt{1792}\): \(1792 = 7 \times 256 = 7 \times 16^2\). So, \(\sqrt{1792} = \sqrt{7 \times 16^2} = \sqrt{16^2} \times \sqrt{7} = 16\sqrt{7}\).

Substituting the Simplified Terms into the Expression for x

Now, substitute these simplified square roots back into the expression for \(x\):

\(x = \left( {\frac{{25\sqrt{3}}}{{36\sqrt{3}}} \div \frac{{20\sqrt{3}}}{{16\sqrt{3}}}} \right) \times \frac{{5\sqrt{7}}}{{16\sqrt{7}}}\)

Calculating the Value of x

Let's simplify the fractions within the expression:

• \(\frac{{25\sqrt{3}}}{{36\sqrt{3}}} = \frac{25}{36}\) (The \(\sqrt{3}\) terms cancel out)
• \(\frac{{20\sqrt{3}}}{{16\sqrt{3}}} = \frac{20}{16} = \frac{5 \times 4}{4 \times 4} = \frac{5}{4}\) (The \(\sqrt{3}\) terms cancel out)
• \(\frac{{5\sqrt{7}}}{{16\sqrt{7}}} = \frac{5}{16}\) (The \(\sqrt{7}\) terms cancel out)

Now substitute these simplified fractions back:

\(x = \left( {\frac{25}{36} \div \frac{5}{4}} \right) \times \frac{5}{16}\)

Perform the division. Remember that dividing by a fraction is the same as multiplying by its reciprocal:

\(\frac{25}{36} \div \frac{5}{4} = \frac{25}{36} \times \frac{4}{5}\)

Now, simplify this multiplication:

\(\frac{25}{36} \times \frac{4}{5} = \frac{5 \times 5}{9 \times 4} \times \frac{4}{5}\)

Cancel out common factors (a 5 from the numerator and denominator, and a 4 from the numerator and denominator):

\(\frac{{\cancel{5} \times 5}}{{9 \times \cancel{4}}} \times \frac{{\cancel{4}}}{{\cancel{5}}} = \frac{5}{9}\)

So, the expression inside the parentheses simplifies to \(\frac{5}{9}\). Now, substitute this back into the expression for \(x\):

\(x = \frac{5}{9} \times \frac{5}{16}\)

Multiply the fractions:

\(x = \frac{5 \times 5}{9 \times 16} = \frac{25}{144}\)

So, the value of \(x\) is \(\frac{25}{144}\).

Finding the Square Root of x

The question asks for the value of \(\sqrt{x}\). We found \(x = \frac{25}{144}\).

\(\sqrt{x} = \sqrt{\frac{25}{144}}\)

Using the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\):

\(\sqrt{\frac{25}{144}} = \frac{\sqrt{25}}{\sqrt{144}}\)

Calculate the square roots:

• \(\sqrt{25} = 5\)
• \(\sqrt{144} = 12\)

So, \(\sqrt{x} = \frac{5}{12}\).

Final Answer

The value of \(\sqrt{x}\) is \(\frac{5}{12}\).

Revision Table: Key Steps for Solving Square Root Problems

Additional Information: Properties of Square Roots and Fractions

Understanding the properties of square roots and fractions is crucial for solving problems like this.

• Product Property of Square Roots: \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\). This is used to simplify roots like \(\sqrt{1875} = \sqrt{625 \times 3} = \sqrt{625} \times \sqrt{3}\).
• Quotient Property of Square Roots: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). This was used at the final step to find \(\sqrt{\frac{25}{144}} = \frac{\sqrt{25}}{\sqrt{144}}\).
• Dividing Fractions: \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\). This rule was applied when evaluating the expression inside the parentheses.
• Multiplying Fractions: \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\). We multiplied the numerators and denominators.
• Cancelling Common Factors: In fractions, if a number appears as a factor in both the numerator and the denominator, it can be cancelled out. For example, \(\frac{20\sqrt{3}}{16\sqrt{3}} = \frac{20}{16}\) because \(\sqrt{3}\) is a common factor. Also, \(\frac{20}{16} = \frac{5 \times 4}{4 \times 4} = \frac{5}{4}\) because 4 is a common factor. This simplifies calculations greatly.

Mastering these fundamental concepts helps in efficiently simplifying complex expressions involving square roots and fractions.