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Precalculus

David Cohen, Theodore B. Lee, David Sklar

Chapter 1

Fundamentals - all with Video Answers

Educators

Section 1

Sets of Real Numbers

02:30

Problem 1

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
(a) $-203$
(b) $203 / 2$

Ayesha Fahmin
Ayesha Fahmin
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01:00

Problem 2

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
(a) $27 / 4$
(b) $\sqrt{27 / 4}$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:54

Problem 3

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
(a) $10^{6}$
(b) $10^{6} / 10^{7}$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:22

Problem 4

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
(a) 8.7
(b) $8 . \overline{7}$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:46

Problem 5

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
(a) 8.74
(b) $8 . \overline{74}$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:51

Problem 6

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
(a) $\sqrt{99}$
(b) $\sqrt{99}+1$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:51

Problem 7

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
$$3 \sqrt{101}+1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:04

Problem 7

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$\pi / 3$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:31

Problem 8

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
$$(3-\sqrt{2})+(3+\sqrt{2})$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:40

Problem 9

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
$$(\sqrt{5}+1) / 4$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:01

Problem 10

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form $\sqrt{n}$, where $n$ is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: $\sqrt{6}$, $\sqrt{10}-2,3 \sqrt{15}, \text { and }-5 \sqrt{3} / 2 .)$
$$(0.1234) /(0.5677)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:41

Problem 11

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$11 / 4$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:49

Problem 12

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$-7 / 8$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:52

Problem 13

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$1+\sqrt{2}$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:58

Problem 14

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$1-\sqrt{2}$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:52

Problem 15

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$\sqrt{2}-1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:58

Problem 16

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$-\sqrt{2}-1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:09

Problem 17

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$\sqrt{2}+\sqrt{3}$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:02

Problem 18

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$\sqrt{2}-\sqrt{3}$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:55

Problem 19

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$(1+\sqrt{2}) / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:01

Problem 20

Draw a number line similar to the one shown in Figure $1 .$ Then indicate the approximate location of the given number. Where necessary, make use of the approximations $\sqrt{2} \approx 1.4$ and $\sqrt{3} \approx 1.7 .$ (The symbol $\approx$ means is approximately equal to.)
$$(2 \sqrt{3}+1) / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:49

Problem 21

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$\pi / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:15

Problem 22

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$3 \pi / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:55

Problem 23

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$\pi / 6$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:25

Problem 24

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$7 \pi / 4$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:29

Problem 25

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$-1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:33

Problem 26

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$3$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:42

Problem 28

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$3 / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:14

Problem 29

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$2 \pi+1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:01

Problem 30

Draw a number line similar to the one shown in Figure $3(a) .$ Then indicate the approximate location of the given number.
$$2 \pi-1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:43

Problem 31

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$-5<-50$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:29

Problem 32

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$0<-1$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:44

Problem 33

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$-2 \leq-2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:39

Problem 34

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$\sqrt{7}-2 \geq 0$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
02:05

Problem 35

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$\frac{13}{14}>\frac{15}{16}$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:43

Problem 36

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$0 . \overline{7}>0.7$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:33

Problem 37

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$2 \pi<6$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:42

Problem 38

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$2 \leq(\pi+1) / 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:34

Problem 39

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$2 \sqrt{2} \geq 2$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:48

Problem 40

Say whether the statement is TRUE or FALSE. (In Exercises $37-40$, do not use a calculator or table; use instead the approximations $\sqrt{2} \approx 1.4$ and $\pi \approx 3.1$ )
$$\pi^{2}<12$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:18

Problem 41

Express each interval using inequality notation and show the given interval on a number line.
$$(2,5)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:05

Problem 42

Express each interval using inequality notation and show the given interval on a number line.
$$(-2,2)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:58

Problem 43

Express each interval using inequality notation and show the given interval on a number line.
$$[1,4]$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:17

Problem 44

Express each interval using inequality notation and show the given interval on a number line.
$$\left[-\frac{3}{2}, \frac{1}{2}\right]$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:09

Problem 45

Express each interval using inequality notation and show the given interval on a number line.
$$[0,3)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:09

Problem 46

Express each interval using inequality notation and show the given interval on a number line.
$$(-4,0]$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:19

Problem 47

Express each interval using inequality notation and show the given interval on a number line.
$$(-3, \infty)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:22

Problem 48

Express each interval using inequality notation and show the given interval on a number line.
$$(\sqrt{2}, \infty)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:55

Problem 49

Express each interval using inequality notation and show the given interval on a number line.
$$[-1, \infty)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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00:55

Problem 50

Express each interval using inequality notation and show the given interval on a number line.
$$[0, \infty)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:56

Problem 51

Express each interval using inequality notation and show the given interval on a number line.
$$(-\infty, 1)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
00:59

Problem 52

Express each interval using inequality notation and show the given interval on a number line.
$$(-\infty,-2)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:19

Problem 53

Express each interval using inequality notation and show the given interval on a number line.
$$(-\infty, \pi]$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
01:19

Problem 54

Express each interval using inequality notation and show the given interval on a number line.
$$(-\infty, \infty)$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
Numerade Educator
02:02

Problem 55

The value of the irrational number $\pi$, correct to ten decimal places (without rounding off), is $3.1415926535 .$ By using a calculator, determine to how many decimal places each of the following quantities agrees with $\pi$.
(a) $(4 / 3)^{4}$ : This is the value used for $\pi$ in the Rhind papyrus, an ancient Babylonian text written about 1650 B.C.
(b) $22 / 7:$ Archimedes $(287-212$ B.C. ) showed that $223 / 71<\pi<22 / 7 .$ The use of the approximation $22 / 7$ for $\pi$ was introduced to the Western world through the writings of Boethius (ca. $480-520$ ), a Roman philosopher, mathematician, and statesman. Among all fractions with numerators and denominators less than $100,$ the fraction $22 / 7$ is the best approximation to $\pi$
(c) $355 / 113:$ This approximation of $\pi$ was obtained in fifth-century China by Zu Chong-Zhi (430-501) and his son. According to David Wells in The Penguin Dictionary of Curious and Interesting Numbers
(Harmondsworth, Middlesex, England: Viking Penguin, Ltd., 1986 ), "This is the best approximation of any fraction below $103993 / 33102$."
(d) $\frac{63}{25}\left(\frac{17+15 \sqrt{5}}{7+15 \sqrt{5}}\right):$ This approximation for $\pi$ was obtained by the Indian mathematician Scrinivasa Ramanujan ( $1887-1920$ ).

Bon Zapata
Bon Zapata
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01:43

Problem 56

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational.
$$a+b$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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01:56

Problem 57

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational.
$$a b$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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02:04

Problem 58

Give an example of irrational numbers a and b such that the indicated expression is (a) rational and (b) irrational.
$$a / b$$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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02:34

Problem 59

(a) Give an example in which the result of raising a rational number to a rational power is an irrational number.
(b) Give an example in which the result of raising an irrational number to a rational power is a rational number.

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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02:55

Problem 60

Can an irrational number raised to an irrational power yield an answer that is rational? This problem shows that the answer is "yes." (However, if you study the following solution very carefully, you'll see that even though we've answered the question in the affirmative, we've not pinpointed the specific case in which an irrational number raised to an irrational power is rational.)
(a) Let $A=(\sqrt{2})^{\sqrt{2}} .$ Now, either $A$ is rational or $A$ is irrational. If $A$ is rational, we are done. Why?
(b) If $A$ is irrational, we are done. Why?
Hint: Consider $A^{\sqrt{2}}$
Remark: For more about this problem and related questions, see the article "Irrational Numbers," by J. P. Jones and S. Toporowski in American Mathematical Monthly, vol. 80( 1973) pp. $423-424$

Rakesh Kumar Sharma
Rakesh Kumar Sharma
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