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Sagot :
Let's go through the questions step by step and provide detailed solutions for each.
### Question 5: Rewrite each mixed radical as an entire radical
#### (a) [tex]\( 3 \sqrt{9} \)[/tex]
To rewrite [tex]\( 3 \sqrt{9} \)[/tex] as an entire radical:
1. Recognize that [tex]\( \sqrt{9} = 3 \)[/tex].
2. Thus, [tex]\( 3 \sqrt{9} = 3 \times 3 = 3^2 \)[/tex].
Converting [tex]\( 3 \sqrt{9} \)[/tex] into an entire radical:
[tex]\[ 3 \sqrt{9} = \sqrt{(3^2 \cdot 9)} = \sqrt{(81)} \][/tex].
The entire radical is [tex]\( 9.0 \)[/tex].
#### (b) [tex]\( -8 \sqrt[3]{9} \)[/tex]
To rewrite [tex]\( -8 \sqrt[3]{9} \)[/tex] as an entire radical:
1. Recognize that [tex]\( \sqrt[3]{9} = 9^{1/3} \)[/tex].
2. Thus, [tex]\( -8 \sqrt[3]{9} = -8 \cdot 9^{1/3} \)[/tex].
Converting [tex]\( -8 \sqrt[3]{9} \)[/tex] into an entire radical:
[tex]\[ -8 \sqrt[3]{9} = (-8) \times (9^{1/3}) = -8 \times 9^{1/3} = (-8^3) \times (9^{3/3}) = (512) \times (9)^{-.33} \][/tex].
The entire radical is [tex]\( -512.0 \)[/tex].
### Question 6: Evaluate each power without using a calculator
#### (a) [tex]\( \frac{64 \cdot 5}{144} \)[/tex]
Evaluate [tex]\( \frac{64 \cdot 5}{144} \)[/tex]:
[tex]\[ \frac{320}{144} = \frac{320 \div 16}{144 \div 16} = \frac{20}{9} \approx 2.2222222222222223 \][/tex].
#### (b) [tex]\( (-27)^{\frac{2}{3}} \)[/tex]
Evaluate [tex]\( (-27)^{\frac{2}{3}} \)[/tex]:
1. Recognize that [tex]\( -27 = (-3)^3 \)[/tex].
2. Thus, [tex]\( (-27)^{\frac{2}{3}} = ((-3)^3)^{\frac{2}{3}} = (-3)^{2} = 9 \)[/tex].
The real component of this evaluation is approximately [tex]\( -4.499999999999997 + 7.794228634059947j \)[/tex], showing the use of complex numbers.
### Question 7: Write each power as a radical
#### (a) [tex]\( 35^{\frac{4}{5}} \)[/tex]
Rewrite [tex]\( 35^{\frac{4}{5}} \)[/tex] as a radical:
[tex]\[ 35^{\frac{4}{5}} = \sqrt[5]{35^4} \][/tex].
The radical form is [tex]\( 17.189151347155786 \)[/tex].
#### (b) [tex]\( \left(\frac{125}{8}\right)^{\frac{2}{3}} \)[/tex]
Rewrite [tex]\( \left(\frac{125}{8}\right)^{\frac{2}{3}} \)[/tex] as a radical:
[tex]\[ \left(\frac{125}{8}\right)^{\frac{2}{3}} = \left( \sqrt[3]{\frac{125}{8}} \right)^2 = \frac{\sqrt[3]{125}^{2}}{\sqrt[3]{8}^{2}} \][/tex].
The radical form is [tex]\( 6.249999999999999 \)[/tex].
### Question 8: Write each radical in exponent form
#### (a) [tex]\( \sqrt[5]{28} \)[/tex]
Write [tex]\( \sqrt[5]{28} \)[/tex] in exponent form:
[tex]\[ \sqrt[5]{28} = 28^{\frac{1}{5}} \][/tex].
#### (b) [tex]\( (\sqrt[4]{16})^{-3} \)[/tex]
Write [tex]\( (\sqrt[4]{16})^{-3} \)[/tex] in exponent form:
[tex]\[ \sqrt[4]{16} = 16^{\frac{1}{4}} \][/tex].
Thus,
[tex]\[ (\sqrt[4]{16})^{-3} = (16^{\frac{1}{4}})^{-3} = 16^{\frac{-3}{4}} = 0.125 \][/tex].
### Question 9: Evaluate each power without using a calculator
#### (a) [tex]\( (64)^{\frac{-2}{3}} \)[/tex]
Evaluate [tex]\( (64)^{\frac{-2}{3}} \)[/tex]:
1. Recognize that [tex]\( 64 = 4^3 \)[/tex].
2. Thus, [tex]\( (64)^{\frac{-2}{3}} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \)[/tex].
The evaluation gives us [tex]\( 0.06250000000000001 \)[/tex].
### Question 5: Rewrite each mixed radical as an entire radical
#### (a) [tex]\( 3 \sqrt{9} \)[/tex]
To rewrite [tex]\( 3 \sqrt{9} \)[/tex] as an entire radical:
1. Recognize that [tex]\( \sqrt{9} = 3 \)[/tex].
2. Thus, [tex]\( 3 \sqrt{9} = 3 \times 3 = 3^2 \)[/tex].
Converting [tex]\( 3 \sqrt{9} \)[/tex] into an entire radical:
[tex]\[ 3 \sqrt{9} = \sqrt{(3^2 \cdot 9)} = \sqrt{(81)} \][/tex].
The entire radical is [tex]\( 9.0 \)[/tex].
#### (b) [tex]\( -8 \sqrt[3]{9} \)[/tex]
To rewrite [tex]\( -8 \sqrt[3]{9} \)[/tex] as an entire radical:
1. Recognize that [tex]\( \sqrt[3]{9} = 9^{1/3} \)[/tex].
2. Thus, [tex]\( -8 \sqrt[3]{9} = -8 \cdot 9^{1/3} \)[/tex].
Converting [tex]\( -8 \sqrt[3]{9} \)[/tex] into an entire radical:
[tex]\[ -8 \sqrt[3]{9} = (-8) \times (9^{1/3}) = -8 \times 9^{1/3} = (-8^3) \times (9^{3/3}) = (512) \times (9)^{-.33} \][/tex].
The entire radical is [tex]\( -512.0 \)[/tex].
### Question 6: Evaluate each power without using a calculator
#### (a) [tex]\( \frac{64 \cdot 5}{144} \)[/tex]
Evaluate [tex]\( \frac{64 \cdot 5}{144} \)[/tex]:
[tex]\[ \frac{320}{144} = \frac{320 \div 16}{144 \div 16} = \frac{20}{9} \approx 2.2222222222222223 \][/tex].
#### (b) [tex]\( (-27)^{\frac{2}{3}} \)[/tex]
Evaluate [tex]\( (-27)^{\frac{2}{3}} \)[/tex]:
1. Recognize that [tex]\( -27 = (-3)^3 \)[/tex].
2. Thus, [tex]\( (-27)^{\frac{2}{3}} = ((-3)^3)^{\frac{2}{3}} = (-3)^{2} = 9 \)[/tex].
The real component of this evaluation is approximately [tex]\( -4.499999999999997 + 7.794228634059947j \)[/tex], showing the use of complex numbers.
### Question 7: Write each power as a radical
#### (a) [tex]\( 35^{\frac{4}{5}} \)[/tex]
Rewrite [tex]\( 35^{\frac{4}{5}} \)[/tex] as a radical:
[tex]\[ 35^{\frac{4}{5}} = \sqrt[5]{35^4} \][/tex].
The radical form is [tex]\( 17.189151347155786 \)[/tex].
#### (b) [tex]\( \left(\frac{125}{8}\right)^{\frac{2}{3}} \)[/tex]
Rewrite [tex]\( \left(\frac{125}{8}\right)^{\frac{2}{3}} \)[/tex] as a radical:
[tex]\[ \left(\frac{125}{8}\right)^{\frac{2}{3}} = \left( \sqrt[3]{\frac{125}{8}} \right)^2 = \frac{\sqrt[3]{125}^{2}}{\sqrt[3]{8}^{2}} \][/tex].
The radical form is [tex]\( 6.249999999999999 \)[/tex].
### Question 8: Write each radical in exponent form
#### (a) [tex]\( \sqrt[5]{28} \)[/tex]
Write [tex]\( \sqrt[5]{28} \)[/tex] in exponent form:
[tex]\[ \sqrt[5]{28} = 28^{\frac{1}{5}} \][/tex].
#### (b) [tex]\( (\sqrt[4]{16})^{-3} \)[/tex]
Write [tex]\( (\sqrt[4]{16})^{-3} \)[/tex] in exponent form:
[tex]\[ \sqrt[4]{16} = 16^{\frac{1}{4}} \][/tex].
Thus,
[tex]\[ (\sqrt[4]{16})^{-3} = (16^{\frac{1}{4}})^{-3} = 16^{\frac{-3}{4}} = 0.125 \][/tex].
### Question 9: Evaluate each power without using a calculator
#### (a) [tex]\( (64)^{\frac{-2}{3}} \)[/tex]
Evaluate [tex]\( (64)^{\frac{-2}{3}} \)[/tex]:
1. Recognize that [tex]\( 64 = 4^3 \)[/tex].
2. Thus, [tex]\( (64)^{\frac{-2}{3}} = 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \)[/tex].
The evaluation gives us [tex]\( 0.06250000000000001 \)[/tex].
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