Answer :
Absolutely! Let's tackle the problem step-by-step.
### Part 1: Converting [tex]\(1.5454\ldots\)[/tex] to a fraction
First, let's express the repeating decimal [tex]\(1.5454\ldots = 1.\overline{54}\)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex].
1. Let [tex]\( x = 1.\overline{54} \)[/tex].
2. To eliminate the repeating part, we multiply [tex]\( x \)[/tex] by 100 (because the repeating block "54" has two digits).
[tex]\[ 100x = 154.\overline{54} \][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 100x - x = 154.\overline{54} - 1.\overline{54} \][/tex]
This simplifies to:
[tex]\[ 99x = 153 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{153}{99} \][/tex]
5. Simplify the fraction [tex]\(\frac{153}{99}\)[/tex]. The greatest common divisor (GCD) of 153 and 99 is 9. Dividing both the numerator and the denominator by 9 gives:
[tex]\[ \frac{153 \div 9}{99 \div 9} = \frac{17}{11} \][/tex]
Therefore, [tex]\( 1.\overline{54} = \frac{17}{11} \)[/tex].
### Part 2(a): Simplify [tex]\((3\sqrt{2} - 2\sqrt{3})^2\)[/tex]
Here, we expand and simplify the expression using the binomial theorem:
1. Start with the square:
[tex]\[ (3\sqrt{2} - 2\sqrt{3})^2 = (3\sqrt{2})^2 - 2 \cdot 3\sqrt{2} \cdot 2\sqrt{3} + (2\sqrt{3})^2 \][/tex]
2. Simplify each term:
[tex]\[ (3\sqrt{2})^2 = 9 \cdot 2 = 18 \][/tex]
[tex]\[ 2 \cdot 3\sqrt{2} \cdot 2\sqrt{3} = 2 \cdot 3 \cdot 2 \cdot \sqrt{2} \cdot \sqrt{3} = 12 \sqrt{6} \][/tex]
[tex]\[ (2\sqrt{3})^2 = 4 \cdot 3 = 12 \][/tex]
3. Combine the terms:
[tex]\[ 18 - 12\sqrt{6} + 12 \][/tex]
4. Summarize:
[tex]\[ 18 + 12 - 12\sqrt{6} = 30 - 12\sqrt{6} \][/tex]
Therefore, [tex]\((3\sqrt{2} - 2\sqrt{3})^2 = 30 - 12\sqrt{6}\)[/tex].
### Part 2(b): Simplify [tex]\((5 + \sqrt{5})(5 - \sqrt{5})\)[/tex]
Here, we recognize the difference of squares:
1. Use the formula [tex]\( (a + b)(a - b) = a^2 - b^2 \)[/tex]:
[tex]\[ (5 + \sqrt{5})(5 - \sqrt{5}) = 5^2 - (\sqrt{5})^2 \][/tex]
2. Simplify each term:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ (\sqrt{5})^2 = 5 \][/tex]
3. Subtract:
[tex]\[ 25 - 5 = 20 \][/tex]
Therefore, [tex]\((5 + \sqrt{5})(5 - \sqrt{5}) = 20\)[/tex].
### Summary of Solutions
1. [tex]\(1.\overline{54} = \frac{17}{11}\)[/tex]
2. [tex]\((3\sqrt{2} - 2\sqrt{3})^2 = 30 - 12\sqrt{6}\)[/tex]
3. [tex]\((5 + \sqrt{5})(5 - \sqrt{5}) = 20\)[/tex]
These are the simplified results of the given mathematical expressions.
### Part 1: Converting [tex]\(1.5454\ldots\)[/tex] to a fraction
First, let's express the repeating decimal [tex]\(1.5454\ldots = 1.\overline{54}\)[/tex] in the form [tex]\(\frac{p}{q}\)[/tex].
1. Let [tex]\( x = 1.\overline{54} \)[/tex].
2. To eliminate the repeating part, we multiply [tex]\( x \)[/tex] by 100 (because the repeating block "54" has two digits).
[tex]\[ 100x = 154.\overline{54} \][/tex]
3. Subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 100x - x = 154.\overline{54} - 1.\overline{54} \][/tex]
This simplifies to:
[tex]\[ 99x = 153 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{153}{99} \][/tex]
5. Simplify the fraction [tex]\(\frac{153}{99}\)[/tex]. The greatest common divisor (GCD) of 153 and 99 is 9. Dividing both the numerator and the denominator by 9 gives:
[tex]\[ \frac{153 \div 9}{99 \div 9} = \frac{17}{11} \][/tex]
Therefore, [tex]\( 1.\overline{54} = \frac{17}{11} \)[/tex].
### Part 2(a): Simplify [tex]\((3\sqrt{2} - 2\sqrt{3})^2\)[/tex]
Here, we expand and simplify the expression using the binomial theorem:
1. Start with the square:
[tex]\[ (3\sqrt{2} - 2\sqrt{3})^2 = (3\sqrt{2})^2 - 2 \cdot 3\sqrt{2} \cdot 2\sqrt{3} + (2\sqrt{3})^2 \][/tex]
2. Simplify each term:
[tex]\[ (3\sqrt{2})^2 = 9 \cdot 2 = 18 \][/tex]
[tex]\[ 2 \cdot 3\sqrt{2} \cdot 2\sqrt{3} = 2 \cdot 3 \cdot 2 \cdot \sqrt{2} \cdot \sqrt{3} = 12 \sqrt{6} \][/tex]
[tex]\[ (2\sqrt{3})^2 = 4 \cdot 3 = 12 \][/tex]
3. Combine the terms:
[tex]\[ 18 - 12\sqrt{6} + 12 \][/tex]
4. Summarize:
[tex]\[ 18 + 12 - 12\sqrt{6} = 30 - 12\sqrt{6} \][/tex]
Therefore, [tex]\((3\sqrt{2} - 2\sqrt{3})^2 = 30 - 12\sqrt{6}\)[/tex].
### Part 2(b): Simplify [tex]\((5 + \sqrt{5})(5 - \sqrt{5})\)[/tex]
Here, we recognize the difference of squares:
1. Use the formula [tex]\( (a + b)(a - b) = a^2 - b^2 \)[/tex]:
[tex]\[ (5 + \sqrt{5})(5 - \sqrt{5}) = 5^2 - (\sqrt{5})^2 \][/tex]
2. Simplify each term:
[tex]\[ 5^2 = 25 \][/tex]
[tex]\[ (\sqrt{5})^2 = 5 \][/tex]
3. Subtract:
[tex]\[ 25 - 5 = 20 \][/tex]
Therefore, [tex]\((5 + \sqrt{5})(5 - \sqrt{5}) = 20\)[/tex].
### Summary of Solutions
1. [tex]\(1.\overline{54} = \frac{17}{11}\)[/tex]
2. [tex]\((3\sqrt{2} - 2\sqrt{3})^2 = 30 - 12\sqrt{6}\)[/tex]
3. [tex]\((5 + \sqrt{5})(5 - \sqrt{5}) = 20\)[/tex]
These are the simplified results of the given mathematical expressions.
