Question

in exercises 53-64, perform the indicated operation and write the expression in simplest form. assume all variables are positive. (see example 7.)

1. $sqrt{81a^{3}b^{5}c^{7}} cdot sqrt{a^{4}b^{7}c^{2}}$
2. $sqrt3{125x^{4}t} cdot sqrt3{x^{3}s^{5}t^{6}}$
3. $sqrt5{16m^{8}n^{4}p} cdot sqrt4{m^{6}np^{7}}$
4. $sqrt5{32x^{2}y^{6}z} cdot sqrt5{x^{9}y^{7}z^{8}}$
5. $c^{3} div sqrt4{d^{7}}$
6. $w^{7} div sqrt5{z^{13}}$
7. $sqrt3{m^{6}} div sqrt3{n^{3}}$
8. $sqrt4{a^{8}} div sqrt4{b^{10}}$
9. $(18w^{1/3}v^{5/4}) div (27w^{4/3}v^{1/2})$
10. $(7x^{3/4}y^{5/2}) div (56x^{-1/2}y^{1/4}z^{-2/3})$
11. $(21x^{5/3}y^{4/5}z^{-2}) div (3x^{1/3}y^{-1/5}z^{-3})$
12. $(36r^{4/3}s^{-1/2}t^{-5/2}) div (4r^{2/3}s^{-2}t^{-5})$

connecting concepts in exercises 65-68, find simplified expressions for the perimeter and area of the given rectangle.

1. rectangle with length $2x^{2/3}$ and width $x^{3}$
2. rectangle with length $4x^{1/3}$ and width $3x^{1/3}$
3. rectangle with length $2sqrt{x}$ and width $sqrt{x}$
4. rectangle with length $3sqrt3{x}$ and width $sqrt3{8x}$

Explanation:

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Exercise 53

Step1: Convert roots to exponents

$\sqrt{81a^3b^5c^7} = 9a^{\frac{3}{2}}b^{\frac{5}{2}}c^{\frac{7}{2}}$, $\sqrt{a^4b^7c^1} = a^2b^{\frac{7}{2}}c^{\frac{1}{2}}$

Step2: Multiply coefficients + add exponents

$9 \cdot 1 \cdot a^{\frac{3}{2}+2}b^{\frac{5}{2}+\frac{7}{2}}c^{\frac{7}{2}+\frac{1}{2}} = 18a^{\frac{7}{2}}b^{6}c^{4}$

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Exercise 54

Step1: Convert roots to exponents

$\sqrt[3]{125r^4t} = 5r^{\frac{4}{3}}t^{\frac{1}{3}}$, $\sqrt[6]{r^3s^5t^0} = r^{\frac{1}{2}}s^{\frac{5}{6}}$

Step2: Multiply coefficients + add exponents

$5 \cdot 1 \cdot r^{\frac{4}{3}+\frac{1}{2}}s^{\frac{5}{6}}t^{\frac{1}{3}} = 5r^{\frac{7}{6}}s^{\frac{13}{6}}t^{\frac{1}{3}}$

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Exercise 55

Step1: Convert roots to exponents

$\sqrt[5]{16m^5n^4p} = 2m n^{\frac{4}{5}}p^{\frac{1}{5}}$, $\sqrt[4]{m^6np^7} = m^{\frac{3}{2}}n^{\frac{1}{4}}p^{\frac{7}{4}}$

Step2: Multiply coefficients + add exponents

$2 \cdot 1 \cdot m^{1+\frac{3}{2}}n^{\frac{4}{5}+\frac{1}{4}}p^{\frac{1}{5}+\frac{7}{4}} = 2m^{\frac{13}{10}}n^{\frac{21}{20}}p^{\frac{9}{20}}$

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Exercise 56

Step1: Convert roots to exponents

$\sqrt[5]{32x^2y^6z} = 2x^{\frac{2}{5}}y^{\frac{6}{5}}z^{\frac{1}{5}}$, $\sqrt[3]{x^9y^7z^{-8}} = x^3y^{\frac{7}{3}}z^{-\frac{8}{3}}$

Step2: Multiply coefficients + add exponents

$2 \cdot 1 \cdot x^{\frac{2}{5}+3}y^{\frac{6}{5}+\frac{7}{3}}z^{\frac{1}{5}-\frac{8}{3}} = 2x^{\frac{19}{15}}y^{\frac{17}{10}}z^{-\frac{37}{15}}$

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Exercise 57

Step1: Convert root to exponent

$\sqrt[4]{d^7} = d^{\frac{7}{4}}$

Step2: Rewrite division as negative exponent

$c^3 \div d^{\frac{7}{4}} = c^3d^{-\frac{7}{4}}$

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Exercise 58

Step1: Convert root to exponent

$\sqrt[5]{z^{13}} = z^{\frac{13}{5}}$

Step2: Rewrite division as negative exponent

$w^7 \div z^{\frac{13}{5}} = w^7z^{-\frac{13}{5}}$

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Exercise 59

Step1: Simplify each cube root

$\sqrt[3]{m^6}=m^2$, $\sqrt[3]{n^3}=n$

Step2: Perform division

$\frac{m^2}{n}$

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Exercise 60

Step1: Simplify each 4th root

$\sqrt[4]{a^8}=a^2$, $\sqrt[4]{b^{10}}=b^{\frac{5}{2}}$

Step2: Perform division

$\frac{a^2}{b^{\frac{5}{2}}}$

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Exercise 61

Step1: Divide coefficients

$\frac{18}{27} = \frac{2}{3}$

Step2: Subtract exponents for like bases

$w^{\frac{1}{3}-\frac{4}{3}}v^{\frac{5}{4}-\frac{1}{2}} = w^{-1}v^{\frac{3}{4}}$

Step3: Combine terms

$\frac{2}{3}w^{-1}v^{\frac{3}{4}} = \frac{2v^{\frac{3}{4}}}{3w}$

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Exercise 62

Step1: Divide coefficients

$\frac{7}{56} = \frac{1}{8}$

Step2: Subtract exponents for like bases

$x^{\frac{3}{4}-(-\frac{1}{2})}y^{\frac{5}{2}-\frac{1}{4}}z^{0-(-\frac{2}{3})} = x y^{\frac{9}{4}-\frac{1}{4}}z^{\frac{2}{3}} = x y z^{\frac{2}{3}}$

Step3: Combine terms

$\frac{1}{8}xyz^{\frac{2}{3}} = \frac{x y z^{\frac{2}{3}}}{8}$

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Exercise 63

Step1: Divide coefficients

$\frac{21}{3} = 7$

Step2: Subtract exponents for like bases

$x^{\frac{5}{3}-\frac{1}{3}}y^{\frac{4}{5}-(-\frac{1}{5})}z^{-2-(-3)} = x^{\frac{4}{3}}y^{\frac{21}{5}}z$

Step3: Combine terms

$7x^{\frac{4}{3}}y^{\frac{21}{5}}z$

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Exercise 64

Step1: Divide coefficients

$\frac{36}{4} = 9$

Step2: Subtract exponents for like bases

$r^{\frac{4}{3}-\frac{2}{3}}s^{-\frac{1}{2}-(-2)}t^{-\frac{5}{2}-(-5)} = r^{\frac{2}{3}}s^{\frac{3}{2}}t^{\frac{5}{2}}$

Step3: Combine terms

$9r^{\frac{2}{3}}s^{\frac{3}{2}}t^{\frac{5}{2}}$

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Exercise 65

Step1: Perimeter of rectangle: $2(l+w)$

$2(2x^{\frac{2}{3}} + x^3) = 4x^{\frac{2}{3}} + 2x^3$

Step2: Area of rectangle: $l \cdot w$

$2x^{\frac{2}{3}} \cdot x^3 = 2x^{\frac{2}{3}+3} = 2x^{\frac{11}{3}}$

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Exercise 66

Step1: Perimeter of rectangle: $2(l+w)$

$2(4x^{\frac{1}{3}} + 3x^{\frac{1}{3}}) = 2(7x^{\frac{1}{3}}) = 14x^{\frac{1}{3}}$

Step2: Area of rectangle: $l \cdot w$

$4x^{\frac{1}{3}} \cdot 3x^{\frac{1}{3}} = 12x^{\frac{2}{3}}$

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Exercise 67

Step1: Perimeter of rectangle: $2(l+w)$

$2(2\sqrt{x} + \sqrt{x}) = 2(3\sqrt{x}) = 6\sqrt{x}$

Step2: Area of rectangle: $l \cdot w$

$2\sqrt{x} \cdot \sqrt{x} = 2x$

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Exercise 68

Step1: Simplify $\sqrt[3]{8x}$

$\sqrt[3]{8x} = 2\sqrt[3]{x}$

Step2: Perimeter of rectangle: $2(l+w)$

$2(3\sqrt[3]{x} + 2\sqrt[3]{x}) = 2(5\sqrt[3]{x}) = 10\sqrt[3]{x}$

Step3: Area of rectangle: $l \cdot w$

$3\sqrt[3]{x} \cdot 2\sqrt[3]{x} = 6x^{\frac{2}{3}}$
*Note: Handwritten area was incorrect; corrected above*

Answer:

Exercises 53-64:

1. $18a^{\frac{7}{2}}b^{6}c^{4}$
2. $5r^{\frac{7}{6}}s^{\frac{13}{6}}t^{\frac{1}{3}}$
3. $2m^{\frac{13}{10}}n^{\frac{21}{20}}p^{\frac{9}{20}}$
4. $2x^{\frac{19}{15}}y^{\frac{17}{10}}z^{-\frac{37}{15}}$
5. $c^3d^{-\frac{7}{4}}$
6. $w^7z^{-\frac{13}{5}}$
7. $\frac{m^2}{n}$
8. $\frac{a^2}{b^{\frac{5}{2}}}$
9. $\frac{2v^{\frac{3}{4}}}{3w}$
10. $\frac{x}{8}y z^{\frac{2}{3}}$
11. $7x^{\frac{4}{3}}y^{\frac{21}{5}}z$
12. $9r^{\frac{2}{3}}s^{\frac{3}{2}}t^{\frac{5}{2}}$

Exercises 65-68:

1. Perimeter: $4x^{\frac{2}{3}} + 2x^3$, Area: $2x^{\frac{11}{3}}$
2. Perimeter: $14x^{\frac{1}{3}}$, Area: $12x^{\frac{2}{3}}$
3. Perimeter: $6\sqrt{x}$, Area: $2x$
4. Perimeter: $2(3\sqrt[3]{x} + \sqrt[3]{8x})$, Area: $3x$