1. Test on Topic 16 Radicals Solutions
Math 99 – Topic 16 Test on Radicals
Test on Topic 16 Radicals Solutions 1.
Evaluate each radical expression, if possible (If it is not a real number, so state.): 36 81 3
 64
(a) 49
(b)
(c) –  4
(d) 1 16
 1
(e) 4
(f) 4
16 = 2
(g) –  25 5
(h) 2.
Evaluate each exponential expression, if possible (If it is not a real number, so state.): 11 4 3 3
(8) 2 3
 1
(a)
(b) 25 2
(c) 3 3.
Express each of the following in their simplest exponential form.
x 
(a) 3
27 x15
(b)
x 7
(c) 5 2 4 5 4.
Simplify the following expressions giving your answer in radical form.
 x 
 4 y  6 4
y13
(a)
x 6
(b) 3
y 17
(c) 5
(d)
(e) 5 3
 16z 
6  2 1
5. Simplify (write the answer in radical form without a negative exponent): 6.
Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers. 3 49 6 5
(a)
(e) 80 64 81
(b)
(f) 20 25 4
(c)
(g)
(d)
(h) 3 32 12 5
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2. Math 99 – Topic 16 Test on Radicals7.
Simplify (Write exact expressions using radicals, not decimals from a calculator).
(a)
(d)
(g) 3
27 x3 y10 z 5
8x 3 y 3 z 5  2x 2 y 3 z
64 xy
8x 7 y 5
(b)
(e)
(h)
2x 4 y 5 z 6  8xy 6 z
64 xy
8x 7 y 5
2 y 3 z 3
(c)
(f)
(i) 3
40x 6 y 5 z 19
72x5 y 4 z 3
36x 3 y 5 w4
8 yz 8.
Simplify by rationalizing the denominator. 12 3
8 y 2 z 5
2 y 36 49
(a)
(b)
(c)
10 x 2 y
40 xy 5 8 2
(d)
8x y z
3 4 5
(e)
(f)
3x 2 y 3
4 xy 4 81 4 81 2 9
10 x 3 y 5
2 xy
(g)
(h)  
(i) 9.
Simplify the following
5  2 5  2 
20  3 2  2 5
(a)
12 + 27
(b)
8 +
(c)
10. Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers. 64 81
8x 3 y 4 z 5 12 3 3 49 36
3x 2 y 3
4 xy 6 5
10 x 2 y
40 xy 5
8 y 2 z 5
2 y 12 5 8 2 10
(a)
(e)
(i)
(b)
(f)
(j)
(c)
(g)
(k)
(d)
(h)
(l)
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3. Math 99 – Topic 16 Test on Radicals
11. Solve the following Radical Equations.
(a)
x = 9
(b)
4x  3 = 9
(c) 2 x = 8
(d)
3x  6 = 3
12. Solve the following Radical Equation.
2x  3 = x
13. Solve the following Radical Equations
2x  17 =
x  7
14. Solve the following Radical Equations.
(a)
4x  5 = 5
(b) x  2  7 = 1
15. Solve the following Radical Equation
2x  17 =
x + 1.
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4. Test on Topic 13 Radicals Solutions
Math 99 – Topic 16 Test on Radicals
Test on Topic 13 Radicals Solutions 1.
Evaluate each radical expression, if possible (If it is not a real number, so state.):
(a) 49 =
7 or – 7 36 81 36 81 6 9 2 3
(b) = = =
(c) –  4 =
Not real number 3
 64
(d) =
– 4 1 16 4 1 1 2 1 2
(e) 4 = =
or – 16
(f) 4
16 = 2
(g) –  25 =
Not a real number 5
 1
(h) =
– 1 2.
Evaluate each exponential expression, if possible (If it is not a real number, so state.):
(8) 2 3
(a) = 3
(8) 2 = 3 64 = 4
 1 1 25 1 25 1 5
(b) 25 2 = = = 1 2 11 4 3 3
11  3 8
(c) = 3 4 4 = 3 4 = = 9 3 3.
Express each of the following in their simplest exponential form. 15
(a) 3
27 x 15 = 3
27  x 3 15 =
3 x 3 =
3x5 7
(b) x 7 = x 2
x 
5 2 .
4 5 10 20 1
x 2 5 2
(c) 4 5 = x = x =
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5.  x   4 y     16z  16 z   16z  1 1 1 4  5 = y13 y 3 4 y 64
Math 99 – Topic 16 Test on Radicals 4.
Simplify the following expressions giving your answer in radical form.
(a)
x 6 = x3
(b) 3
y 17 =
y 5 3 y 2
 x 
 52  6 6
 x 
(c) 5 =
  =
x15   4
y13
y y
(d) =
 4 y 
  5 3
(e)
43 y5 3
64 y15 =
64  y15 =
8 y y = =
 16z 
6  2 1
5. Simplify (write the answer in radical form without a negative exponent):
 16z 
6  2 1 1
6 2 1
16 z 6 1
16 z 6 1
4 z 3 = = = =
16 z  6.
Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers. 80
16  5 =
16  5
(a) = =
4 5
(b)
(c) 20 3 49 =
4  5 3 49 =
4  5 = 3 7
2 5
(d) 3 32 = 3
8  4 = 3
8  3 4 =
23 4 64 81 25 4 64 81 25 4 8 9 5 2
(e)
(f) = = 6 5
6  5
5  5 30 5
(f) = = 12 5 12 5
12  5
5  5 60 5
4 15 5
4  15 5
2 15 5
(g) = = = = = =
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6. Math 99 – Topic 16 Test on Radicals7.
Simplify (Write exact expressions using radicals, not decimals from a calculator).
(b) 3
27 x3 y10 z 5 = 3
27  3 x3  3 y10  3 z 5
= 3xy3 y z 3 z 2 3 =
3xy3z 3 yz 2
(c)
2x 4 y 5 z 6  8xy 6 z = =
2x 4 y 5 z 6  8xy 6 z
16x5 y11z 7
16  x5  y11  z 7
4x2 x y5 y z3 z
4x2y5z3 xyz
(c) 3
40x 6 y 5 z 19 = 3
40  3 x 6  3 y 5  3 z 19 = 3
8  5  3 y 5  3 z 19
23 5  x2  y  z 6  3 z
2x2 yz z =
(d)
8x 3 y 3 z 5  2x 2 y 3 z =
16x 5 y 6 z 6 =
16 x 5
y z 6 = 4x2 x y3z3 = 4x2y3z3 x
64 xy
8x 7 y 5 8 6 8
x6  y 4
2 2
x3 y 2
(e) = = =
x y 4
72x5 y 4 z 3
72  x5  y 4  z3
6 2  x2 x  y 2  z z = 6x2 y xz
(f) = =
64 xy
8x 7 y 5 8 6 8
x6  y 4
2 2
x3 y 2
(g) = = =
x y 4
2 y 3 z 3
2 y 3 z
8 yz 3
y 2
4 z 2
y 2
4  z y
2 z
(h) = = = =
8 yz 2
(i)
36x3 y5 w4  36  x3  y5  w4  6  x x  y y  w2  6xy 2 w2 xy
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7. Math 99 – Topic 16 Test on Radicals8.
Simplify by rationalizing the denominator. 12 3 12 3 3
12 3 3
(a) =  = =
4 3
8 y 2 z 5
2 y
8 y 2 z 5
2 y
2 y
8 y 2 z y
2 y
(b) =  = =
4 yz y 36 49 36 49 6 7
(c) = =
(d)
8x 3 y 4 z 5 =
8 x 3
y z 5 =
2 2 x x y 2 z z =
2xy 2 z 2xz
10 x 2 y
40 xy 5 x
4 y 2 x
4 y 2 x
2 y
(e) = = = 8 2 8 2  2
8 2 2
(f) = = =
4 2
3x 2 y 3
4 xy
3x 2 y 3
4 xy
4 xy
3 x 2 y xy
4 xy
3 xy x y 4
6 xy xy 4
3 xy xy 2
(g) =  = = = = 4 81
(h)  4 81  2 9
10 x3 y 5
2 xy
10 x3 y5
2 xy
2 xy
10 x3 y xy
2 xy
5 x 2 y xy 1
(i)    
 5x2 y xy 9.
Simplify the following
(a)
12 + 27 =
2 3  3 3  5 3
(b)
8 +
20  3 2  2 5  2 2  2 5  3 2  2 5   2
5  2 5  2   25  5
(c)
2  5 2  2  3
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8. Math 99 – Topic 16 Test on Radicals
10. Simplify (Write exact expressions using radicals, not decimals from a calculator).
Assume the variables represent positive numbers. 3 49 64 81 3 49 64 81 3 7 8 9
(a)
(b) = = 6 5
6  5
5  5 30 5
(c) = = 12 5 12 5
12  5
5  5 60 5
4 15 5
4  15 5
2 15 5
(d) = = = = = = 36 49 36 49 6 7
(e) = =
(f)
8x 3 y 4 z 5 =
8 x 3
y z 5 =
2 2 x x y 2 z z =
2xy 2 z 2xz
10 x 2 y
40 xy 5 x
4 y 2 x
4 y 2 x
2 y
(g) = = = 8 2 8 2 2
8 2 2
(h) =  = =
4 2
3x 2 y 3
4 xy
3x 2 y 3
4 xy
4 xy
3x 2 y xy
4 xy
3xy x y 4
6 xy xy 4
3xy xy 2
(i) =  = = = = 12 3 12 3 3
12 3 3
(j) =  = =
4 3
8 y 2 z 5
2 y
8 y 2 z 5
2 y
8 y 2 z y
2 y
(k) = 
2 y = =
4 yz y 5 10 5 10 10
5 10 10 10 2
(l) =  = =
11. Solve the following Radical Equations.
(a)
x =
4x  3 = 9
(b) 9 x = 81
4x – 3 = 81 4x x = 84 21
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9. Math 99 – Topic 16 Test on Radicals
2 x x = 8 4 16
(d)
3x  6 = 3
3x – 6
3x =
= 9 15 x
12. Solve the following Radical Equation. = 5
2x  3
2x + 3 = x x2
x2 – 2x – 3
(x – 3)(x + 1)
x = 3 or x = – 1
but after checking x = – 1 is unusable its a “phantom solution” so only x = 3 works
13. Solve the following Radical Equations.
2x  17
2x + 17 2x x
x  7
x – 7
x – 24
– 24 =
(after check x = – 24 is not a usable solution so there is no solution to this equation)
14. Solve the following Radical Equations.
4x  5 =
(b) x  2  7
(a) 5 = 1
2 3x  2
4x + 5 = 25 = 8
3x  2
4x = 20 = 4 x = 5
3x + 2 = 16 3x x = 14
15. Solve the following Radical Equation
2x  17 =
x + 1.
2x  17
2x + 17 17 =
x+1
(x +1)2
(x + 1)(x + 1)
x2 + 2x + 1
x2 + 1
x2 – 16
(x + 4)(x – 4)
So x = – 4 or x = 4 but after checking x = – 4 is unusable so only x = 4 is a solution
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