11 ALGEBRA

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Университет имени Сулеймана Демиреля

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15. Evaluate

C) 2ñ6 – 2

D) 8 + 5ñ3

E) 2 + 2ñ6

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D. RADICAL EQUATIONS

Definition

An equation that has a variable in a radicand is called a radical equation.

For example, the equations ñx = 25, x 1 3, and 2x – 1 3x 5 are radical equations. Let us look at some examples of how to solve radical equations.

EXAMPLE 19 Solve x 1 3.

Solution ( x 1)2 32 (take the square of both sides) x 1 9

x 8

	20 Solve		3x 1 5.
EXAMPLE			3x 1 5.

Solution (		3x 1)2 52
		3x 1 25
			3x 24
			x 8
	21 Solve		2x 3 3 8.
EXAMPLE			2x 3 3 8.

Solution		2x 3 3		8
			2x 3	8 – 3
			2x 3	5
	(	2x 3)2 52
			2x 3 25
			2x 22
			x 11

Check: 8 1 3

9 3

3 3 Therefore, 8 is a solution.

Check: 3 8 1 5

25 5

5 5 Therefore, 8 is a solution.

Check:	?
	2 11 3 3 8

25 3 8

5 3 8

8 8 Therefore, 11 is a solution.

Radical Functions

EXAMPLE 22 Solve x2 12 x 6.

Solution ( x2 12)2 (x 6)2

x2 12 x2 12x 36 12x –24

x –2

EXAMPLE 23 Solve 3 3x 2 5.

Solution (3 3x 2)3 53

3x 2 125

3x 123 x 41

	24 Solve 4x 1 –	5x – 1 0.
EXAMPLE		5x – 1 0.

Solution 4x 1 – 5x – 1		0
	( 4x 1)2	( 5x – 1)2

4x 1 5x – 1 x 2

Check: (–2)2 12 – 2 6

4 12 4

16 4

4 4

Therefore, –2 is a solution.

Check: 3 3 41 2 5

3 123 2 5

3 125 5

5 5

Therefore, 41 is a solution.

Check:	4 2 – 1 –	?
		5 2 – 1 0

9 – 9 0

3 – 3 0

0 0

Therefore, 2 is a solution.

100	Algebra 11

	25 Solve 3x 1			3x 6 5.
EXAMPLE				3x 6 5.

Solution		3x 1	5 –		3x 6
(		3x 1)2	(5 –		3x 6)2 (take the square of both sides)
		3x 1	25 – 2 5			3x 6	3x 6	(simplify)
	10	3x 6	30
(		3x 6)2	32
		3x 6	9

3x 3
x 1
Check: 3 1 1 –	3 1	?
Check: 3 1 1 –	3 1	6 5
	4	?
	4	9 5

2 3 5

5 5

Therefore, 1 is a solution.

Check Yourself 5

Solve each equation and check your answer.

a. ñx = 5		b. ñx + 1 = 3
d. ñx = –3		e.	2x – 1 7
g.	3x – 5 4 6	h.	5x 1 – 3 0
			2
j.	4x – 3 – 2x – 2 0	k.	x 3 x – 2 5
l.	x2 1 2 – x	m. 3 x 1 3
o. 3 3x 2 3 5x		p. 4 x 4 2x – 1

c.óx+1 = 6

f. 3 x 1 – 1 8

2x 6 – x

n. 3 x 2

Radical Functions

101

EXERCISES 2.1

A.Radical Expressions

1.Evaluate the expressions.

a.	4	16	b.	4 16	c.	4 16
d.	3	27	e.	25	f.	25 3
		8		4
g.	3	27 2	h.	4 0.0625	i.	7 128

2.Write each expression in its simplest form. All the variables represent positive real numbers.


a. 4x4 y12		b.	3 25a6 y12
c.	4 64x4 y20	d.	5 32x15 y20
e.	5 4 xy	f.	3 3	3 x2 y3
g.	4 x16	h. 3 9a2 3 9a
i.	4 x 3 y5	j.	12 243x6 y12
				xy

k. 16 312(x+ y)12		l.	4 3	(x a)20(a x)28

3. Write each expression in its simplest form. All the variables represent positive real numbers.

a. 3 x2 6 x5

b.	x5	4 x3
	x 3 x2
c.	(x2	xy 3 y2 )2 4 x3

d. 4 58 x 4 34 x5 + 4 252 x9

e. 4 2x 4 x3 4 4x2

2xy 2x 2y

4.Write each expression in its simplest form.

a.3 16 3 54 + 3 128

b.4 32 + 4 162 4 512

c.4 162 3 54 + 4 32 + 3 16

5. Rationalize the denominator of each fraction.

5 2

3 2

3 5

12 x11 1

3 2 1

x +1

3 x

3 3 3 2

4 2 1

3 16 3 12 + 3 9

(16 3 +1)(

8 3 +1)( 4 3 +1)(

3 +1)

6.Evaluate each expression.

a.56+ 56+ 56+...

b.6 32 6 32 6 32 ...

c.4 32 4 32 4 32 ...

102	Algebra 11

7. Prove each equality.

a.	n a n a n a ... = n 1 a
b.	n a n a n a ... = n 1 a
c.	n a m a n a m a ... = mn 1 am 1
d.	a+ a+ a+... =	4a+1+1

	2

B.Rational Exponents

8.Evaluate each expression.

a.	91/2	b.	–91/2	c.	(–9)1/2
d.	–81/3	e.	(–8)1/3	f.	01/16
g.	163/4	h.	324/5

9. For each statement, give a possible value for a.

a.	(a2)1/2 a	b.	(a2)1/2	= a
c.	(a2)2 = a	d.	(a1/2)2	a

e. (a1/2)2 = a

10.Evaluate the expressions.

a. (–8)7/3 91/2

b. 641/3 + 31/2 9 3–5/2

c. 41.5 ( 91) 0.5 ( 641 ) 2 / 3 ( 45)3.5 0.83.5

11.Simplify the expressions. All the variables represent positive real numbers.

a. (2x1/2) (3x1/3)		b. x2/5 251/2 x3/5
c.	4x5 / 2 1/ 2	d.	1/2		1/2		1/2	1/2
	( x2 / 5 )		(x – y )(x + y )
e.	361/ 2 x1/ 4	f.		–1/2	y	2/5	3
	641/ 2 x3 / 5		(2x				)

12.Write each expression as a radical expression. All the variables represent positive real numbers.

a.	a3/5	b.	x–1/3	c.	a–4/5
d.	x–1/2y1/2	e.	(x1/2)1/3	f.	x–2/7
g.	y–5/3	h.	5a3/2	i.	(x2y3)1/2
j.	(4ab2)5/7	k.	(7x3y)–1/3	l.	(a + b)3/2

13.Write each expression as an expression with a rational exponent. All the variables represent positive real numbers.

a.	3 x	b.	4 a
c.	x2 + y2	d. 5 (3a)2
e.	3 (x y) 1	f.	6 x12 y12
g. ña + ñb		h. 34 a3
i.	5x5 y2	j.	7 (x2 y3 )3
k.	9 (x 3y)2	l.	3 a + 5 x+ y2

Radical Functions

103

14.Simplify each expression by converting the radicals into rational powers.

a.	4 a2	3 a
		a
b.	c4 / 5 3 a 2		(8a 2c) 1/ 3
b.	5	c 1	(8a 2c) 1/ 3
	5	c 1

15.Simplify each expression.

a.(x1/2 + 2y1/2)(x1/2 – 3y1/2)

b.(3a1/2 – b1/2)(a1/2 – 2b1/2)

c.2x1/3(3x2/3 – x8/3)

d.3a4/5(2a1/5 – a6/5)

e.(3x1/2 – 2y1/2)(3x1/2 + 2y1/2)

f.(3x1/2 + y1/2)2

16. Simplify each expression.


a.	a2 / 5	b2 / 5
	a1/ 5	b1/ 5
b.		a3 / 4 +b3			/ 4
b.	a1/ 2	a1/ 4 b1/ 4			+ b1/ 2
c.	x+ y1/ 2			x4 / 3		y2 / 3		1/ 3 1/ 6
							1/ 3 + x y
	1/ 3	1/ 6			2 / 3		1/ 3 + x y
	x	+ y		x		+ y
d.	c 27		3c1/ 3 +18
	c2 / 3 9			c1/ 3 +3

C.Exponential Equations

17.Solve the equations.

a.32x = 81

b.7x = 72 – x

c.25x = 53 – x

d.3x2 – 5x +8 = 9

18.(23)x = 4 1.5 is given. Find x.

19.Find the real number y which satisfies

x= 22y+4 and x3 = 210y+4.

20.Find the real numbers x and y which satisfy 0.0064y 1000x = 45.

21.Find the real number x which satisfies

 (–4x + 1)2n = (x2 + 2x + 1)n, n .

22.Find the possible values of a which satisfy  2a+1 + 4a = 80.

23.Find the real number x which satisfies

xm 15 + xm 16 + xm 17

xm 18 + xm 17 + xm 16 =128.

104	Algebra 11

D.Radical Equations

24.Write the expressions in exponential form and simplify if possible.

a.	ò21	b.	72	c.	7 5	d. 3 a2 x3
e.	3 5 2	f.	( 5 3)6	g.	3 1
					3

25. Write the expressions in radical form.

	1		2			3	–	3		ab			3		1
a.	a2	b. b		c.	(				d.	x c	e.	(a		)
			3			2)		2					2		6

26. Simplify the expressions.
a.		3 3		b.	4	5 a2	c.	1016
d.	3	5	5	e.	5 243		f.	(–5)4
g.	3	–27		h. 3 64x6			i.	a3 3 a2
j.	3	2 3 4		k.	4	3 4 27	l.	162
								2
m.		3	2	n.	4	3 21	o.	3
m.	3 250			n.	6	7	o.	3 5
p.	3 34 9 3			q. 4 23 3 24 25

27. Perform the operations.

a.		3 3 2	b.	2 3 3 6 6
		6 108
c.	3	11– 3 4 – 3 27	d.	7	3	2	10
c.			d.	2 9		2	27
e. 4 17 6 8			f.	( 2 – 6)( 2 3 )

28. Solve the equations.

a.	2x 1 3		b.	3x – 14 8
c. ò7x = x			d.	3 4x 6
e. 3 5x – 4 6			f.	3 7x 6 5
g.	5x – 1 3 7		h.	2x – 3 – 2 4
i.	x2 5 x 1		j.	x2 9 5
k. 3 7x – 6 4			l.	2x – 5 x – 4
m. x		x 16 – 2	n.	6	2
				x	3
o. ñ6x – x= 5
p.	x 1 2 x x 1– 2 x 8

q.33x – 6 1

92x 1 93

r.	4 3 8x – 2	6			1
				4x – 3

s.	1	–	1			2

	2 – x		2		x

Radical Functions

105

Mixed Problems

29. Evaluate

a 3 / 2 b(ab–2 )–1/ 2	(a–1 )–2 / 3	3	if a =	2	and b =	1	.

			2			3 2

30. Evaluate	4 8	2 1	4 8	2 1 .
		4 8	2 1

31. Order the numbers 16 64, 10 74 7 and 4 2	5 .
	4


32. Find the sum of all values of x which satisfy  (x – 3)x2 – 6x + 5 = 1.

33. Find the possible values of x which satisfy



x 25 x 25 x 25 ... = 25.

34. Solve the equation (7 3 5x 6 )+90 = 3 5x .



35. Solve the equation	3		x 3		10	7 = 0, x .
35. Solve the equation		5		+ 3 5x 3
		5		+ 3 5x 3

36. Solve the equation x 2 2x = 4.



37. Solve the equation



3	5 + 3+ 5 =10 x+2 2 x+1.

106	Algebra 11

CHAPTER REVIEW TEST 2A

1. Evaluate 43 82		163.
A) 212	B) 224	C) 424	D) 812	E) 168

2. Solve 9x+2 = 243 for x.
A)	1	B)	1	C) 1	D) 2	E) 3
	3		2

3. Simplify 2k 7k (–2)–k if k is an odd integer.

A) 14k B) 27k C) 7–k D) 7k E) (–7)k

4.a = (84)7, b = (45)8 and c = 2(34) are given. Which one of the following is correct?

A) a < b < c	B) c < b < a
C) c < a < b	D) b < c < a
	E) b < a < c

5. Evaluate 810.25 + 490.5 + 320.2.
A) 10	B) 12	C) 14	D) 16	E) 20

6. Simplify 23a – b 22b – a + 2a – b 2a + 2b.

A) 22b+a+1	B) 2b – 2a	C) 22a+b+1
D) 22a – b		E) 22a+2

7.	Simplify	(2x+1 )3
7.	Simplify	(2	x+2 2 .
		(2	)
	A) 2x – 1		B) 2x+1	C) 22x+1
		D) 2x – 2		E) 22x+2

8. Simplify	23t +23t +23t +23t +23t +23t +23t +23t .
			8t+1
A) 0	B) 1	C) 2	D) 2t	E) 2t+1

Chapter Review Test 2A

107

9. Evaluate 100–3 58	45.
A) 200 B) 400	C) 50 D) 100 E) 250

10. Evaluate	4+	23+ 4 .
A) 3	B) 4	C) 5	D) 6	E) 8

11. Evaluate	0.64 +	0.09 2	2.25.
A) –2.4	B) –2.2	C) –1.9	D) –1.1 E) –0.9

12. Simplify	18k5	72k3 .
A) 6k4	B) 36k4	C) ñ6k D) k3	E) 36k

13. Evaluate		3	+	2+	1 .
		8			4
A)	3				B)	3		C) ò18
A)	2				B)	2		C) ò18
	2					2
	D)	5					E)	15
	D)	4					E)	8
		4						8

14. Evaluate 5 3 0.008.

A) 3 0.04 B) 3 0.2 C) 0.01 D) 0.1 E) 1

16. Order 5ñ2, 4ñ3 and 7.
A) 4ñ3 < 5ñ2 < 7	B) 5ñ2 < 7 < 4ñ3
C) 7 < 4ñ3 < 5ñ2	D) 4ñ3 < 7 < 5ñ2

E) 7 < 5ñ2 < 4ñ3

108	Algebra 11

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