Question

1. evaluate ( 81^{\frac{1}{4}} ). (lesson 7 - 4)
2. simplify ( sqrt{98} ). (lesson 7 - 5)
3. simplify ( sqrt{121a^{5}b^{5}c} ). (lesson 7 - 5)

a. ( 11a^{2}bsqrt{abc} )
b. ( 11a^{2}b^{4} )
c. ( 11absqrt{ab^{3}c} )
d. ( 11bsqrt{a^{5}b^{5}c} )

1. simplify ( sqrt3{\frac{125a^{7}}{27b^{3}}} ). (lesson 7 - 5)

a. ( \frac{a^{2}}{b} )
b. ( \frac{5a^{6}}{3b^{3}} )
c. ( \frac{sqrt3{a^{7}}}{sqrt3{b}} )
d. ( \frac{5a^{2}}{3b} )

1. simplify ( 8sqrt{5}-6sqrt{5} ). (lesson 7 - 6)
2. simplify ( (8t^{6}u^{9})(3w^{7}) ). (lesson 7 - 4)

a. ( 11t^{6}u^{24} )
b. ( 24t^{6}u^{24} )
c. ( 24t^{6}u^{7} )
d. ( 24t^{6}u^{16} )

1. simplify ( (cd^{3})^{\frac{1}{3}} ). (lesson 7 - 4)

a. ( cd^{\frac{1}{3}} )
b. ( c^{\frac{1}{3}}d^{\frac{1}{3}} )
c. ( c^{\frac{1}{3}}d^{9} )
d. ( cd^{9} )

1. simplify ( 6sqrt3{4}cdot8sqrt3{28} ). (lesson 7 - 5)

a. ( 14sqrt3{32} )
b. ( 48sqrt3{112} )
c. ( 48sqrt3{14} )
d. ( 96sqrt3{14} )

1. simplify ( 4sqrt3{7}+9sqrt3{56} ). (lesson 7 - 6)

a. ( 4sqrt3{7}+9sqrt3{56} )
b. ( 13sqrt3{7} )
c. ( 22sqrt3{7} )
d. ( 36sqrt3{7} )

Explanation:

25. Step1: Rewrite as square root

$81^{\frac{1}{2}} = \sqrt{81}$

25. Step2: Evaluate the square root

$\sqrt{81} = 9$

26. Step1: Factor 98 into perfect square

$\sqrt{98} = \sqrt{49 \times 2}$

26. Step2: Simplify the radical

$\sqrt{49 \times 2} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}$

27. Step1: Split the radical into factors

$\sqrt{121a^5b^3c} = \sqrt{121} \times \sqrt{a^4 \times a} \times \sqrt{b^2 \times b} \times \sqrt{c}$

27. Step2: Simplify each radical term

$\sqrt{121} \times a^2\sqrt{a} \times b\sqrt{b} \times \sqrt{c} = 11a^2b\sqrt{abc}$

28. Step1: Split cube root into fractions

$\sqrt[3]{\frac{125a^6}{27b^3}} = \frac{\sqrt[3]{125a^6}}{\sqrt[3]{27b^3}}$

28. Step2: Simplify numerator and denominator

$\frac{\sqrt[3]{125} \times \sqrt[3]{a^6}}{\sqrt[3]{27} \times \sqrt[3]{b^3}} = \frac{5a^2}{3b}$

29. Step1: Combine like radical terms

$8\sqrt{5} - 6\sqrt{5} = (8-6)\sqrt{5}$

29. Step2: Calculate the coefficient

$(8-6)\sqrt{5} = 2\sqrt{5}$

30. Step1: Multiply coefficients and variables

$(8t^2u^5)(3tu^5) = (8 \times 3) \times (t^2 \times t) \times (u^5 \times u^5)$

30. Step2: Simplify using exponent rules

$24 \times t^{2+1} \times u^{5+5} = 24t^3u^{10}$

31. Step1: Apply power of a power rule

$(cd^{\frac{2}{3}})^3 = c^{1 \times 3} \times d^{\frac{2}{3} \times 3}$

31. Step2: Simplify exponents

$c^3d^2$

32. Step1: Multiply coefficients and radicals

$6\sqrt[3]{4} \cdot 8\sqrt[3]{28} = (6 \times 8) \times \sqrt[3]{4 \times 28}$

32. Step2: Simplify the product inside radical

$48 \times \sqrt[3]{112} = 48 \times \sqrt[3]{8 \times 14}$

32. Step3: Simplify the radical term

$48 \times \sqrt[3]{8} \times \sqrt[3]{14} = 48 \times 2 \times \sqrt[3]{14} = 96\sqrt[3]{14}$

33. Step1: Rewrite $\sqrt[3]{56}$ as factored radical

$9\sqrt[3]{56} = 9\sqrt[3]{8 \times 7} = 9 \times 2\sqrt[3]{7} = 18\sqrt[3]{7}$

33. Step2: Combine like radical terms

$4\sqrt[3]{7} + 18\sqrt[3]{7} = (4+18)\sqrt[3]{7}$

33. Step3: Calculate the coefficient

$(4+18)\sqrt[3]{7} = 22\sqrt[3]{7}$

Answer:

1. $9$
2. $7\sqrt{2}$
3. A. $11a^2b\sqrt{abc}$
4. D. $\frac{5a^2}{3b}$
5. $2\sqrt{5}$
6. D. $24t^3u^{10}$
7. B. $c^3d^2$
8. D. $96\sqrt[3]{14}$
9. C. $22\sqrt[3]{7}$