Answer :
Let's go through the steps to rationalize the denominator for each expression given and express the decimals in fraction form:
45. Rationalize the denominator of the expression [tex]\(\frac{30}{5 \sqrt{3}-3 \sqrt{5}}\)[/tex]:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator [tex]\(5 \sqrt{3} + 3 \sqrt{5}\)[/tex]:
[tex]\[ \frac{30}{5 \sqrt{3}-3 \sqrt{5}} \times \frac{5 \sqrt{3}+3 \sqrt{5}}{5 \sqrt{3}+3 \sqrt{5}} \][/tex]
The denominator thus becomes:
[tex]\[ (5 \sqrt{3})^2 - (3 \sqrt{5})^2 = 25 \cdot 3 - 9 \cdot 5 = 75 - 45 = 30 \][/tex]
The numerator becomes:
[tex]\[ 30 \times (5 \sqrt{3} + 3 \sqrt{5}) = 30 \times 5 \sqrt{3} + 30 \times 3 \sqrt{5} = 150 \sqrt{3} + 90 \sqrt{5} \][/tex]
Thus, the rationalized expression is:
[tex]\[ \frac{150 \sqrt{3} + 90 \sqrt{5}}{30} = 5 \sqrt{3} + 3 \sqrt{5} \approx 15.368457970343757 \][/tex]
A6. Rationalize the denominator of the expression [tex]\(\frac{1}{2+\sqrt{3}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator [tex]\(2-\sqrt{3}\)[/tex]:
[tex]\[ \frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} \][/tex]
The denominator thus becomes:
[tex]\[ (2)^2 - (\sqrt{3})^2 = 4 - 3 = 1 \][/tex]
The numerator becomes:
[tex]\[ 1 \cdot (2-\sqrt{3}) = 2 - \sqrt{3} \][/tex]
Therefore, the rationalized expression is:
[tex]\[ 2 - \sqrt{3} \][/tex]
47. Rationalize the denominator of the expression [tex]\(\frac{16}{\sqrt{4T}-5}\)[/tex]:
Multiply both the numerator and denominator by the conjugate of the denominator [tex]\(\sqrt{4T} + 5\)[/tex]:
[tex]\[ \frac{16}{\sqrt{4T}-5} \times \frac{\sqrt{4T}+5}{\sqrt{4T}+5} \][/tex]
The denominator becomes:
[tex]\[ (\sqrt{4T})^2 - (5)^2 = 4T - 25 \][/tex]
The numerator becomes:
[tex]\[ 16 \times (\sqrt{4T} + 5) = 16\sqrt{4T} + 80 \][/tex]
Therefore, the rationalized expression is:
[tex]\[ \frac{16\sqrt{4T} + 80}{4T - 25} \][/tex]
48. Express the number 2.576 in the [tex]\(\frac{p}{q}\)[/tex] form, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]:
To express 2.576 as a fraction:
[tex]\[ 2.576 = \frac{2576}{1000} \][/tex]
Simplifying this fraction:
[tex]\[ \frac{2576 \div 8}{1000 \div 8} = \frac{322}{125} \][/tex]
Thus, [tex]\( \frac{322}{125} \)[/tex] is the required fraction.
49. Express the number 3.6\overline{8} in the [tex]\(\frac{p}{q}\)[/tex] form, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]:
Let [tex]\( x = 3.688888\ldots \)[/tex]
Then multiply by 10 to shift the repeating part:
[tex]\[ 10x = 36.88888\ldots \][/tex]
Next, subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 10x - x = 36.88888\ldots - 3.68888\ldots \][/tex]
[tex]\[ 9x = 33.2 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{33.2}{9} \][/tex]
Multiplying numerator and denominator by 10 to eliminate the decimal in the numerator:
[tex]\[ x = \frac{332}{90} = \frac{166}{45} \][/tex]
Thus, [tex]\(3.6\overline{8} = \frac{166}{45} \)[/tex].
That completes the step-by-step solutions for rationalizing denominators and converting decimals to fractions.
45. Rationalize the denominator of the expression [tex]\(\frac{30}{5 \sqrt{3}-3 \sqrt{5}}\)[/tex]:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator [tex]\(5 \sqrt{3} + 3 \sqrt{5}\)[/tex]:
[tex]\[ \frac{30}{5 \sqrt{3}-3 \sqrt{5}} \times \frac{5 \sqrt{3}+3 \sqrt{5}}{5 \sqrt{3}+3 \sqrt{5}} \][/tex]
The denominator thus becomes:
[tex]\[ (5 \sqrt{3})^2 - (3 \sqrt{5})^2 = 25 \cdot 3 - 9 \cdot 5 = 75 - 45 = 30 \][/tex]
The numerator becomes:
[tex]\[ 30 \times (5 \sqrt{3} + 3 \sqrt{5}) = 30 \times 5 \sqrt{3} + 30 \times 3 \sqrt{5} = 150 \sqrt{3} + 90 \sqrt{5} \][/tex]
Thus, the rationalized expression is:
[tex]\[ \frac{150 \sqrt{3} + 90 \sqrt{5}}{30} = 5 \sqrt{3} + 3 \sqrt{5} \approx 15.368457970343757 \][/tex]
A6. Rationalize the denominator of the expression [tex]\(\frac{1}{2+\sqrt{3}}\)[/tex]:
Multiply the numerator and the denominator by the conjugate of the denominator [tex]\(2-\sqrt{3}\)[/tex]:
[tex]\[ \frac{1}{2+\sqrt{3}} \times \frac{2-\sqrt{3}}{2-\sqrt{3}} \][/tex]
The denominator thus becomes:
[tex]\[ (2)^2 - (\sqrt{3})^2 = 4 - 3 = 1 \][/tex]
The numerator becomes:
[tex]\[ 1 \cdot (2-\sqrt{3}) = 2 - \sqrt{3} \][/tex]
Therefore, the rationalized expression is:
[tex]\[ 2 - \sqrt{3} \][/tex]
47. Rationalize the denominator of the expression [tex]\(\frac{16}{\sqrt{4T}-5}\)[/tex]:
Multiply both the numerator and denominator by the conjugate of the denominator [tex]\(\sqrt{4T} + 5\)[/tex]:
[tex]\[ \frac{16}{\sqrt{4T}-5} \times \frac{\sqrt{4T}+5}{\sqrt{4T}+5} \][/tex]
The denominator becomes:
[tex]\[ (\sqrt{4T})^2 - (5)^2 = 4T - 25 \][/tex]
The numerator becomes:
[tex]\[ 16 \times (\sqrt{4T} + 5) = 16\sqrt{4T} + 80 \][/tex]
Therefore, the rationalized expression is:
[tex]\[ \frac{16\sqrt{4T} + 80}{4T - 25} \][/tex]
48. Express the number 2.576 in the [tex]\(\frac{p}{q}\)[/tex] form, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]:
To express 2.576 as a fraction:
[tex]\[ 2.576 = \frac{2576}{1000} \][/tex]
Simplifying this fraction:
[tex]\[ \frac{2576 \div 8}{1000 \div 8} = \frac{322}{125} \][/tex]
Thus, [tex]\( \frac{322}{125} \)[/tex] is the required fraction.
49. Express the number 3.6\overline{8} in the [tex]\(\frac{p}{q}\)[/tex] form, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]:
Let [tex]\( x = 3.688888\ldots \)[/tex]
Then multiply by 10 to shift the repeating part:
[tex]\[ 10x = 36.88888\ldots \][/tex]
Next, subtract the original [tex]\( x \)[/tex] from this equation:
[tex]\[ 10x - x = 36.88888\ldots - 3.68888\ldots \][/tex]
[tex]\[ 9x = 33.2 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{33.2}{9} \][/tex]
Multiplying numerator and denominator by 10 to eliminate the decimal in the numerator:
[tex]\[ x = \frac{332}{90} = \frac{166}{45} \][/tex]
Thus, [tex]\(3.6\overline{8} = \frac{166}{45} \)[/tex].
That completes the step-by-step solutions for rationalizing denominators and converting decimals to fractions.
