9.9: Exercise Supplement

Exercise Supplement

Square Root Expressions - Addition and Subtraction of Square Root Expressions

For the following problems, simplify the expressions.

Exercise $\PageIndex{1}$

$\sqrt{10}\sqrt{2}$

Answer

$2\sqrt{5}$

Exercise $\PageIndex{2}$

$\sqrt{6}\sqrt{8}$

Exercise $\PageIndex{3}$

$\sqrt{18}\sqrt{40}$

Answer

$12\sqrt{5}$

Exercise $\PageIndex{4}$

$\sqrt{11}\sqrt{11}$

Exercise $\PageIndex{5}$

$\sqrt{y}\sqrt{y}$

Answer

$y$

Exercise $\PageIndex{6}$

$\sqrt{r^3}\sqrt{r^3}$

Exercise $\PageIndex{7}$

$\sqrt{m+3}\sqrt{m+3}$

Answer

$m+3$

Exercise $\PageIndex{8}$

$\sqrt{a-7}\sqrt{a-7}$

Exercise $\PageIndex{9}$

$\sqrt{x^2+4x+4}$

Answer

$x+2$

Exercise $\PageIndex{10}$

$\sqrt{y^2 - 12y + 36}$

Exercise $\PageIndex{11}$

$\dfrac{\sqrt{x+5}}{\sqrt{x+2}}$

Answer

$\dfrac{\sqrt{(x+5)(x+2)}}{x+2}$

Exercise $\PageIndex{12}$

$\dfrac{\sqrt{n-3}}{\sqrt{n-1}}$

Exercise $\PageIndex{13}$

$\dfrac{\sqrt{50}}{\sqrt{2}}$

Answer

$5$

Exercise $\PageIndex{14}$

$\dfrac{\sqrt{75}}{5\sqrt{3}}$

Exercise $\PageIndex{15}$

$\dfrac{\sqrt{a^2 + 6a + 9}}{\sqrt{a + 3}}$

Answer

$\sqrt{a+3}$

Exercise $\PageIndex{16}$

$\dfrac{\sqrt{4x^2 + 4x + 1}}{\sqrt{2x + 1}}$

Exercise $\PageIndex{17}$

$\dfrac{\sqrt{x^2 - 11x + 24}}{\sqrt{x-8}}$

Answer

$\sqrt{x-3}$

Exercise $\PageIndex{18}$

$\dfrac{\sqrt{y^2 + 11y + 28}}{\sqrt{y+4}}$

Exercise $\PageIndex{19}$

$\sqrt{3}(\sqrt{5} + \sqrt{3})$

Answer

$3 + \sqrt{15}$

Exercise $\PageIndex{20}$

$\sqrt{5}(\sqrt{6} - \sqrt{10})$

Exercise $\PageIndex{21}$

$\sqrt{a}(\sqrt{a} - \sqrt{bc})$

Answer

$a - \sqrt{abc}$

Exercise $\PageIndex{22}$

$\sqrt{x}(\sqrt{x^5} - \sqrt{3x})$

Exercise $\PageIndex{23}$

$\sqrt{7a^3}(\sqrt{2a} - \sqrt{4a^3})$

Answer

$a^2\sqrt{14} - 2a^3\sqrt{7}$

Exercise $\PageIndex{24}$

$\dfrac{3}{\sqrt{7}}$

Exercise $\PageIndex{25}$

$\dfrac{2}{\sqrt{5}}$

Answer

$\dfrac{2\sqrt{5}}{5}$

Exercise $\PageIndex{26}$

$\dfrac{6}{\sqrt{2}}$

Exercise $\PageIndex{27}$

$\dfrac{8y}{\sqrt{y}}$

Answer

$8\sqrt{y}$

Exercise $\PageIndex{28}$

$\dfrac{16a^2}{\sqrt{5a}}$

Exercise $\PageIndex{29}$

$(2 + \sqrt{3})(2 - \sqrt{3})$

Answer

$1$

Exercise $\PageIndex{30}$

$(x + \sqrt{8})(3x + \sqrt{8})$

Exercise $\PageIndex{31}$

$(4y - \sqrt{3x})(4y + \sqrt{3x})$

Answer

$16y^2 - 3x$

Exercise $\PageIndex{32}$

$(6r + \sqrt{2s})(4r + \sqrt{2s})$

Exercise $\PageIndex{33}$

$\dfrac{2}{2 + \sqrt{7}}$

Answer

$-\dfrac{2(2- \sqrt{7})}{3}$

Exercise $\PageIndex{34}$

$\dfrac{4}{1 - \sqrt{6}}$

Exercise $\PageIndex{35}$

$\dfrac{6}{x + \sqrt{y}}$

Answer

$\dfrac{6(x - \sqrt{y})}{x^2 - y}$

Exercise $\PageIndex{36}$

$\dfrac{10}{a - \sqrt{2b}}$

Exercise $\PageIndex{37}$

$\dfrac{\sqrt{5}}{a + \sqrt{3}}$

Answer

$\dfrac{a\sqrt{5} - \sqrt{15}}{a^2 - 3}$

Exercise $\PageIndex{38}$

$\dfrac{\sqrt{2}}{1 + \sqrt{10}}$

Exercise $\PageIndex{39}$

$\dfrac{8 + \sqrt{3}}{2 + \sqrt{6}}$

Answer

$\dfrac{8\sqrt{6} - 2\sqrt{3} + 3\sqrt{2} - 16}{2}$

Exercise $\PageIndex{40}$

$\dfrac{4 + \sqrt{11}}{4 - \sqrt{11}}$

Exercise $\PageIndex{41}$

$\sqrt{\dfrac{36a^4b^5c^{11}}{x^2y^5}}$

Answer

$\dfrac{6a^2b^2c^5\sqrt{bcy}}{xy^3}$

Exercise $\PageIndex{42}$

$\sqrt{x^{12}y^{10}z^8w^7}$

Exercise $\PageIndex{43}$

$\sqrt{32x^5y(x-2)^3}$

Answer

$4x^2(x-2)\sqrt{2xy(x-2)}$

Exercise $\PageIndex{44}$

$-2\sqrt{60r^4s^3}$

Exercise $\PageIndex{45}$

$\sqrt{\dfrac{3}{16}}$

Answer

$\dfrac{\sqrt{3}}{4}$

Exercise $\PageIndex{46}$

$\sqrt{\dfrac{4}{25}}$

Exercise $\PageIndex{47}$

$\sqrt{\dfrac{9}{16}}$

Answer

$\dfrac{3}{4}$

Exercise $\PageIndex{48}$

$\sqrt{\dfrac{5}{36}}$

Exercise $\PageIndex{49}$

$\sqrt{\dfrac{1}{6}}$

Answer

$\dfrac{\sqrt{6}}{6}$

Exercise $\PageIndex{50}$

$\sqrt{\dfrac{3}{10}}$

Exercise $\PageIndex{51}$

$\sqrt{(x+4)^4(x-1)^5}$

Answer

$(x+4)^2(x-1)^2(\sqrt{x-1})$

Exercise $\PageIndex{52}$

$\sqrt{(3x + 5)^3(2x - 7)^3}$

Exercise $\PageIndex{53}$

$\sqrt{(y-3z)^{12}(y+3z)^{10}(y-5z)^3}$

Answer

$(y-3z)^6(y+3z)^5(y-5z)\sqrt{y-5z}$

Exercise $\PageIndex{54}$

$\sqrt{(8a-5b)^{26}(2a - 9b)^{40}(a-b)^{15}}$

Exercise $\PageIndex{55}$

$4\sqrt{11} + 8\sqrt{11}$

Answer

$12\sqrt{11}$

Exercise $\PageIndex{56}$

$-\sqrt{6} + 5\sqrt{6}$

Exercise $\PageIndex{57}$

$5\sqrt{60} - 7\sqrt{15}$

Answer

$3\sqrt{15}$

Exercise $\PageIndex{58}$

$4ax^2\sqrt{75x^4} + 6a\sqrt{3x^8}$

Exercise $\PageIndex{59}$

$-3\sqrt{54} - 16\sqrt{96}$

Answer

$-73\sqrt{6}$

Exercise $\PageIndex{60}$

$\sqrt{18x^2y}\sqrt{2x^2y}$

Exercise $\PageIndex{61}$

$\sqrt{4x^2+32x+64} + \sqrt{10x^2+80x+160}$

Answer

$(2 + \sqrt{10})(x + 4)$

Exercise $\PageIndex{62}$

$-2\sqrt{9x^2 - 42x + 49} + 5\sqrt{18x^2 - 84x + 98}$

Exercise $\PageIndex{63}$

$-10\sqrt{56a^3b^7} + 2a^2b\sqrt{126ab^5}$

Answer

$(-20ab^3 + 6a^2b^3)\sqrt{14ab}$

Exercise $\PageIndex{64}$

$\dfrac{\sqrt{3x} - \sqrt{5x}}{\sqrt{7x} + \sqrt{2x}}$

Exercise $\PageIndex{65}$

$\dfrac{\sqrt{6a} + \sqrt{2a}}{\sqrt{3a} - \sqrt{5a}}$

Answer

$\dfrac{-3\sqrt{2} - \sqrt{30} - \sqrt{6} - \sqrt{10}}{2}$

Square Root Equations with Applications

For the following problems, solve the equations.

Exercise $\PageIndex{66}$

$\sqrt{3x} = 9$

Exercise $\PageIndex{67}$

$\sqrt{4a} = 16$

Answer

$a = 64$

Exercise $\PageIndex{68}$

$\sqrt{x} + 7 = 4$

Exercise $\PageIndex{69}$

$\sqrt{a + 6} = -5$

Answer

No Solution

Exercise $\PageIndex{70}$

$\sqrt{4a + 5} = 21$

Exercise $\PageIndex{71}$

$\sqrt{3m + 7} = 10$

Answer

$m = 31$

Exercise $\PageIndex{72}$

$\sqrt{y + 10} = 5$

Exercise $\PageIndex{73}$

$\sqrt{a - 7} = 6$

Answer

$a = 43$

Exercise $\PageIndex{74}$

$\sqrt{4x - 8} = x - 2$

Exercise $\PageIndex{75}$

$\sqrt{2x + 3} + 8 =11$

Answer

$x=3$

Exercise $\PageIndex{76}$

$\sqrt{a^2 + 5} + 5 = a$

Exercise $\PageIndex{77}$

$\sqrt{5b + 4} - 5 = -2$

Answer

$b = 1$

Exercise $\PageIndex{78}$

$\sqrt{2a + 1} - 10 = -3$

Exercise $\PageIndex{79}$

$\sqrt{2x + 5} = \sqrt{x + 3}$

Answer

$x = -2$

Exercise $\PageIndex{80}$

$\sqrt{5a - 11}$

Exercise $\PageIndex{81}$

At a small business, the monthly number of sales $S$ is approximately related to the number of employees $E$ by $S = 140 + 8\sqrt{E - 2}$

a) Determine the approximate number of sales if the number of employees is $27$.

b) Determine the approximate number of employees if the monthly sales are 268.

Answer

a) $S = 180$

b) $E = 258$

Exercise $\PageIndex{82}$

The resonance frequency $f$ in an electronic circuit containing inductance $L$ and capacitance $C$ in series is given by:

$f = \dfrac{1}{2\pi\sqrt{LC}}$

a) Determine the resonance frequency in an electronic circuit if the inductance is $9$ and the capacitance is $0.0001$. Use $\pi = 3.14$.

b) Determine the inductance in an electric circuit if the resonance frequency is $5.308$ and the capacitance is $0.0001$. Use $\pi = 3.14$.

Exercise $\PageIndex{83}$

If two magnetic poles of strength $m$ and $m'$ are at a distance $r$ centimeters (cm) apart, the force \9F\) of repulsion in air between them is given by:

$F = \dfrac{mm'}{r^2}$

a) Determine the force of repulsion if two magnetic poles of strengths 22 and 46 units are 8 cm apart.

b) Determine how far apart are two magnetic poles of strengths 14 and 16 units if the force of repulsion in air between them is 42 units.

Answer

a) $F=15.8125$  

b) $r=12.31$ cm