Answer :
Alright, let's go step-by-step through each question based on the true numerical results provided.
### Question 09
Evaluate [tex]\(\left(5^{-2}\right)^3\)[/tex]:
1. [tex]\(5^{-2}\)[/tex] means [tex]\(\frac{1}{5^2} = \frac{1}{25}\)[/tex].
2. Raising this to the power of 3 gives us [tex]\(\left(\frac{1}{25}\right)^3 = \frac{1}{25^3}\)[/tex].
3. [tex]\(25^3 = (5^2)^3 = 5^6\)[/tex].
So, [tex]\(\left(5^{-2}\right)^3 = \frac{1}{5^6}\)[/tex].
Thus, the correct answer is:
A. [tex]\(\frac{1}{5^6}\)[/tex].
### Question 10
Determine the type of number [tex]\(\frac{\sqrt{50}}{\sqrt{98}}\)[/tex]:
1. [tex]\(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\)[/tex].
2. [tex]\(\sqrt{98} = \sqrt{2 \times 7^2} = 7\sqrt{2}\)[/tex].
3. Simplify [tex]\(\frac{5\sqrt{2}}{7\sqrt{2}} = \frac{5}{7}\)[/tex].
[tex]\(\frac{5}{7}\)[/tex] is a rational number.
Thus, the correct answer is:
D. Rational
### Question 11
Evaluate [tex]\((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})\)[/tex]:
1. This is a difference of squares: [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Here, [tex]\(a = \sqrt{5}\)[/tex] and [tex]\(b = \sqrt{3}\)[/tex].
3. So, [tex]\((\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2\)[/tex].
Thus, the correct answer is:
C. 2
### Question 12
Convert [tex]\(4.1 \overline{8} 5\)[/tex] to a fraction:
1. [tex]\(4.1\overline{8}5\)[/tex] represents a repeating decimal.
2. It's equivalent to [tex]\(\frac{4185}{999}\)[/tex].
Thus, the correct answer is:
C. [tex]\(\frac{4185}{999}\)[/tex]
### Question 13
Convert [tex]\(0.\overline{3}\)[/tex] to a fraction:
1. [tex]\(0.\overline{3}\)[/tex] means [tex]\(0.3333\ldots\)[/tex], which is equivalent to [tex]\(\frac{1}{3}\)[/tex].
Thus, the correct answer is:
A. [tex]\(\frac{1}{3}\)[/tex]
### Question 14
Evaluate [tex]\(\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{6}\)[/tex]:
1. Combining the radicals: [tex]\(\sqrt{2 \cdot 3 \cdot 6} = \sqrt{36} = 6\)[/tex].
Thus, the value is:
6.0
### Question 15
Determine the type of decimal expansion for [tex]\(\frac{47}{50}\)[/tex]:
1. [tex]\(\frac{47}{50} = 0.94\)[/tex], which is a finite decimal expansion.
Thus, it is a finite decimal.
### Question 16
Evaluate [tex]\((729)^{\frac{1}{3}}\)[/tex]:
1. The cube root of 729 is 9 (since [tex]\(9^3 = 729\)[/tex]).
Thus, the value is:
9.0
### Question 17
Evaluate [tex]\({}^{11} \sqrt{1}\)[/tex]:
1. The 11th root of 1 is simply 1 (since [tex]\(1^{1/11} = 1\)[/tex]).
Thus, the value is:
1.0
### Question 18
Determine the type of decimal expansion for [tex]\(\frac{2}{3}\)[/tex]:
1. [tex]\(\frac{2}{3}\)[/tex] represents the repeating decimal [tex]\(0.\overline{6}\)[/tex].
Thus, it is a repeating decimal.
### Question 19
Evaluate [tex]\((64)^{-\frac{1}{6}}\)[/tex]:
1. First, evaluate [tex]\(64^{\frac{1}{6}}\)[/tex].
2. The sixth root of 64 is 2 (since [tex]\(2^6 = 64\)[/tex]).
3. Therefore, [tex]\((64)^{-\frac{1}{6}} = \frac{1}{2}\)[/tex].
Thus, the value is:
0.5
### Question 09
Evaluate [tex]\(\left(5^{-2}\right)^3\)[/tex]:
1. [tex]\(5^{-2}\)[/tex] means [tex]\(\frac{1}{5^2} = \frac{1}{25}\)[/tex].
2. Raising this to the power of 3 gives us [tex]\(\left(\frac{1}{25}\right)^3 = \frac{1}{25^3}\)[/tex].
3. [tex]\(25^3 = (5^2)^3 = 5^6\)[/tex].
So, [tex]\(\left(5^{-2}\right)^3 = \frac{1}{5^6}\)[/tex].
Thus, the correct answer is:
A. [tex]\(\frac{1}{5^6}\)[/tex].
### Question 10
Determine the type of number [tex]\(\frac{\sqrt{50}}{\sqrt{98}}\)[/tex]:
1. [tex]\(\sqrt{50} = \sqrt{2 \times 5^2} = 5\sqrt{2}\)[/tex].
2. [tex]\(\sqrt{98} = \sqrt{2 \times 7^2} = 7\sqrt{2}\)[/tex].
3. Simplify [tex]\(\frac{5\sqrt{2}}{7\sqrt{2}} = \frac{5}{7}\)[/tex].
[tex]\(\frac{5}{7}\)[/tex] is a rational number.
Thus, the correct answer is:
D. Rational
### Question 11
Evaluate [tex]\((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})\)[/tex]:
1. This is a difference of squares: [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
2. Here, [tex]\(a = \sqrt{5}\)[/tex] and [tex]\(b = \sqrt{3}\)[/tex].
3. So, [tex]\((\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2\)[/tex].
Thus, the correct answer is:
C. 2
### Question 12
Convert [tex]\(4.1 \overline{8} 5\)[/tex] to a fraction:
1. [tex]\(4.1\overline{8}5\)[/tex] represents a repeating decimal.
2. It's equivalent to [tex]\(\frac{4185}{999}\)[/tex].
Thus, the correct answer is:
C. [tex]\(\frac{4185}{999}\)[/tex]
### Question 13
Convert [tex]\(0.\overline{3}\)[/tex] to a fraction:
1. [tex]\(0.\overline{3}\)[/tex] means [tex]\(0.3333\ldots\)[/tex], which is equivalent to [tex]\(\frac{1}{3}\)[/tex].
Thus, the correct answer is:
A. [tex]\(\frac{1}{3}\)[/tex]
### Question 14
Evaluate [tex]\(\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{6}\)[/tex]:
1. Combining the radicals: [tex]\(\sqrt{2 \cdot 3 \cdot 6} = \sqrt{36} = 6\)[/tex].
Thus, the value is:
6.0
### Question 15
Determine the type of decimal expansion for [tex]\(\frac{47}{50}\)[/tex]:
1. [tex]\(\frac{47}{50} = 0.94\)[/tex], which is a finite decimal expansion.
Thus, it is a finite decimal.
### Question 16
Evaluate [tex]\((729)^{\frac{1}{3}}\)[/tex]:
1. The cube root of 729 is 9 (since [tex]\(9^3 = 729\)[/tex]).
Thus, the value is:
9.0
### Question 17
Evaluate [tex]\({}^{11} \sqrt{1}\)[/tex]:
1. The 11th root of 1 is simply 1 (since [tex]\(1^{1/11} = 1\)[/tex]).
Thus, the value is:
1.0
### Question 18
Determine the type of decimal expansion for [tex]\(\frac{2}{3}\)[/tex]:
1. [tex]\(\frac{2}{3}\)[/tex] represents the repeating decimal [tex]\(0.\overline{6}\)[/tex].
Thus, it is a repeating decimal.
### Question 19
Evaluate [tex]\((64)^{-\frac{1}{6}}\)[/tex]:
1. First, evaluate [tex]\(64^{\frac{1}{6}}\)[/tex].
2. The sixth root of 64 is 2 (since [tex]\(2^6 = 64\)[/tex]).
3. Therefore, [tex]\((64)^{-\frac{1}{6}} = \frac{1}{2}\)[/tex].
Thus, the value is:
0.5
