9.9: Exercise Supplement
Exercise Supplement
Square Root Expressions - Addition and Subtraction of Square Root Expressions
For the following problems, simplify the expressions.
\(\sqrt{10}\sqrt{2}\)
- Answer
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\(2\sqrt{5}\)
\(\sqrt{6}\sqrt{8}\)
\(\sqrt{18}\sqrt{40}\)
- Answer
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\(12\sqrt{5}\)
\(\sqrt{11}\sqrt{11}\)
\(\sqrt{y}\sqrt{y}\)
- Answer
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\(y\)
\(\sqrt{r^3}\sqrt{r^3}\)
\(\sqrt{m+3}\sqrt{m+3}\)
- Answer
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\(m+3\)
\(\sqrt{a-7}\sqrt{a-7}\)
\(\sqrt{x^2+4x+4}\)
- Answer
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\(x+2\)
\(\sqrt{y^2 - 12y + 36}\)
\(\dfrac{\sqrt{x+5}}{\sqrt{x+2}}\)
- Answer
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\(\dfrac{\sqrt{(x+5)(x+2)}}{x+2}\)
\(\dfrac{\sqrt{n-3}}{\sqrt{n-1}}\)
\(\dfrac{\sqrt{50}}{\sqrt{2}}\)
- Answer
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\(5\)
\(\dfrac{\sqrt{75}}{5\sqrt{3}}\)
\(\dfrac{\sqrt{a^2 + 6a + 9}}{\sqrt{a + 3}}\)
- Answer
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\(\sqrt{a+3}\)
\(\dfrac{\sqrt{4x^2 + 4x + 1}}{\sqrt{2x + 1}}\)
\(\dfrac{\sqrt{x^2 - 11x + 24}}{\sqrt{x-8}}\)
- Answer
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\(\sqrt{x-3}\)
\(\dfrac{\sqrt{y^2 + 11y + 28}}{\sqrt{y+4}}\)
\(\sqrt{3}(\sqrt{5} + \sqrt{3})\)
- Answer
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\(3 + \sqrt{15}\)
\(\sqrt{5}(\sqrt{6} - \sqrt{10})\)
\(\sqrt{a}(\sqrt{a} - \sqrt{bc})\)
- Answer
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\(a - \sqrt{abc}\)
\(\sqrt{x}(\sqrt{x^5} - \sqrt{3x})\)
\(\sqrt{7a^3}(\sqrt{2a} - \sqrt{4a^3})\)
- Answer
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\(a^2\sqrt{14} - 2a^3\sqrt{7}\)
\(\dfrac{3}{\sqrt{7}}\)
\(\dfrac{2}{\sqrt{5}}\)
- Answer
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\(\dfrac{2\sqrt{5}}{5}\)
\(\dfrac{6}{\sqrt{2}}\)
\(\dfrac{8y}{\sqrt{y}}\)
- Answer
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\(8\sqrt{y}\)
\(\dfrac{16a^2}{\sqrt{5a}}\)
\((2 + \sqrt{3})(2 - \sqrt{3})\)
- Answer
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\(1\)
\((x + \sqrt{8})(3x + \sqrt{8})\)
\((4y - \sqrt{3x})(4y + \sqrt{3x})\)
- Answer
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\(16y^2 - 3x\)
\((6r + \sqrt{2s})(4r + \sqrt{2s})\)
\(\dfrac{2}{2 + \sqrt{7}}\)
- Answer
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\(-\dfrac{2(2- \sqrt{7})}{3}\)
\(\dfrac{4}{1 - \sqrt{6}}\)
\(\dfrac{6}{x + \sqrt{y}}\)
- Answer
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\(\dfrac{6(x - \sqrt{y})}{x^2 - y}\)
\(\dfrac{10}{a - \sqrt{2b}}\)
\(\dfrac{\sqrt{5}}{a + \sqrt{3}}\)
- Answer
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\(\dfrac{a\sqrt{5} - \sqrt{15}}{a^2 - 3}\)
\(\dfrac{\sqrt{2}}{1 + \sqrt{10}}\)
\(\dfrac{8 + \sqrt{3}}{2 + \sqrt{6}}\)
- Answer
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\(\dfrac{8\sqrt{6} - 2\sqrt{3} + 3\sqrt{2} - 16}{2}\)
\(\dfrac{4 + \sqrt{11}}{4 - \sqrt{11}}\)
\(\sqrt{\dfrac{36a^4b^5c^{11}}{x^2y^5}}\)
- Answer
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\(\dfrac{6a^2b^2c^5\sqrt{bcy}}{xy^3}\)
\(\sqrt{x^{12}y^{10}z^8w^7}\)
\(\sqrt{32x^5y(x-2)^3}\)
- Answer
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\(4x^2(x-2)\sqrt{2xy(x-2)}\)
\(-2\sqrt{60r^4s^3}\)
\(\sqrt{\dfrac{3}{16}}\)
- Answer
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\(\dfrac{\sqrt{3}}{4}\)
\(\sqrt{\dfrac{4}{25}}\)
\(\sqrt{\dfrac{9}{16}}\)
- Answer
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\(\dfrac{3}{4}\)
\(\sqrt{\dfrac{5}{36}}\)
\(\sqrt{\dfrac{1}{6}}\)
- Answer
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\(\dfrac{\sqrt{6}}{6}\)
\(\sqrt{\dfrac{3}{10}}\)
\(\sqrt{(x+4)^4(x-1)^5}\)
- Answer
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\((x+4)^2(x-1)^2(\sqrt{x-1})\)
\(\sqrt{(3x + 5)^3(2x - 7)^3}\)
\(\sqrt{(y-3z)^{12}(y+3z)^{10}(y-5z)^3}\)
- Answer
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\((y-3z)^6(y+3z)^5(y-5z)\sqrt{y-5z}\)
\(\sqrt{(8a-5b)^{26}(2a - 9b)^{40}(a-b)^{15}}\)
\(4\sqrt{11} + 8\sqrt{11}\)
- Answer
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\(12\sqrt{11}\)
\(-\sqrt{6} + 5\sqrt{6}\)
\(5\sqrt{60} - 7\sqrt{15}\)
- Answer
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\(3\sqrt{15}\)
\(4ax^2\sqrt{75x^4} + 6a\sqrt{3x^8}\)
\(-3\sqrt{54} - 16\sqrt{96}\)
- Answer
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\(-73\sqrt{6}\)
\(\sqrt{18x^2y}\sqrt{2x^2y}\)
\(\sqrt{4x^2+32x+64} + \sqrt{10x^2+80x+160}\)
- Answer
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\((2 + \sqrt{10})(x + 4)\)
\(-2\sqrt{9x^2 - 42x + 49} + 5\sqrt{18x^2 - 84x + 98}\)
\(-10\sqrt{56a^3b^7} + 2a^2b\sqrt{126ab^5}\)
- Answer
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\((-20ab^3 + 6a^2b^3)\sqrt{14ab}\)
\(\dfrac{\sqrt{3x} - \sqrt{5x}}{\sqrt{7x} + \sqrt{2x}}\)
\(\dfrac{\sqrt{6a} + \sqrt{2a}}{\sqrt{3a} - \sqrt{5a}}\)
- Answer
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\(\dfrac{-3\sqrt{2} - \sqrt{30} - \sqrt{6} - \sqrt{10}}{2}\)
Square Root Equations with Applications
For the following problems, solve the equations.
\(\sqrt{3x} = 9\)
\(\sqrt{4a} = 16\)
- Answer
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\(a = 64\)
\(\sqrt{x} + 7 = 4\)
\(\sqrt{a + 6} = -5\)
- Answer
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No Solution
\(\sqrt{4a + 5} = 21\)
\(\sqrt{3m + 7} = 10\)
- Answer
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\(m = 31\)
\(\sqrt{y + 10} = 5\)
\(\sqrt{a - 7} = 6\)
- Answer
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\(a = 43\)
\(\sqrt{4x - 8} = x - 2\)
\(\sqrt{2x + 3} + 8 =11\)
- Answer
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\(x=3\)
\(\sqrt{a^2 + 5} + 5 = a\)
\(\sqrt{5b + 4} - 5 = -2\)
- Answer
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\(b = 1\)
\(\sqrt{2a + 1} - 10 = -3\)
\(\sqrt{2x + 5} = \sqrt{x + 3}\)
- Answer
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\(x = -2\)
\(\sqrt{5a - 11}\)
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
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a) \(S = 180\)
b) \(E = 258\)
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 \(\pi = 3.14\).
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
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a) \(F=15.8125\)
b) \(r=12.31\) cm