HP 9s Instructions for use

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HP 9s Powers and Roots
Powers and Roots
Practice Solving Problems Involving Powers and Roots

hp calculators
HP 9s Powers and Roots
Powers and roots
The number ain amis said to be raised to the power m, which is also called index or exponent. It obeys the so-called
index laws, namely:
mnnm
nmnm
nmnm
a)a(
aaa
aaa
=
=÷
=×
−
+
where ais any number and mand nare rational numbers. The process of raising to a power is often referred to as
involution. The opposite of a power is called a root. For example, if 53= 125 then 5 is the third root of 125, and it is often
written thus: 31255 =
The process of finding a root is known as extraction and also as evolution. Evolution is the inverse of the involution.
Notice the following important relationships:
n
naa =
1and n
na
a1 =
−
The HP 9s has numerous functions to calculate powers and roots. These are: O, ~i, the exponential function
~h, ~o (same as 1/x), the power function B, Fwhich calculates square roots (i.e. second roots),
~f, that calculates cube roots (the third root), and the xth-root function ~b.
Practice solving problems involving powers and roots
Example 1: Is greater than ?
π
ee π
Solution: We’ll calculate the difference
π
−π e
e
~aB1~h-1~hB~a\
The Band ~h keys are postfix functions, i.e. their argument is the number being displayed. They
do not cancel any pending operation (unless they give an error, e.g. 231~h, which is an
overflow condition). Because of the lower priority of the -key, when it is pressed the partial result is
displayed (22.45915772). The \key at the end carries out all pending operations and returns the final
result.
e
π
We can save one keystroke by storing (i.e. pressing {) the value of ein the M register as soon as it’s
displayed. ecan now be retrieved by pressing one key: [(instead of 1~h):
~aB1~h{-[B~a\
Press 0{ or U{ to clear the M register. (The sequence given in the manual, J{, won’t
work here because there’s no entry to be cleared by Jonce \has been pressed!).
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hp calculators
HP 9s Powers and Roots
The difference is –0.681534914 = π
−π e
e< 0, therefore π
e> e π.Answer:
2: Example e catheti d (s figure 1).Find the hypotenuse of a triangle th of which are 8 an 15 ee
Solution: The hypotenuse of a right triangle is given by Pythagoras’ theorem (even
though the Babylonians already knew this relationship!):
22 baHypotenuse +=
where abare the two catheti. In this example:
he addition. If the \key had been pressed at the end instead, the
and
8O+15O\F
The \key is necessary to perform t
calculation performed would have been 22 158 +. In fact, we can avoid pressing the \key:
M8O+15ONF
is one keystroke shorter, you might prefer using parentheses
for being clearer.
Answer:
Even though the sequence with the \key
mbers 8, 15 and 17 is an example of a Pythagorean triple, i.e. integers that can be the
sides of the same right triangle. Some of the simpler sets were already known by the ancient Egyptians.)
ple 3:
17 . (The set of nu
Refer to the HP 9s learning module Polar/Rectangular Coordinate Conversions to learn a faster way of
calculating the hypotenuse.
Calculate 0 0Exam
Solution: Although s eom calculators return 1, on the HP 9s 0B0\ is an error condition because is
mathematically an indeterminate (or undetermined) form, much like 0 / 0 or log 0. Press Uto remove
xample 4:
0
0
the error indicator.
Ese the exponential function to confirm the result.Calculate 270 9.−U
Solution: A convenie of computing y xis as xlny
esince xlny eex y==nt way The HP-35, the world’s first
scientific electronic pocket calc s ethod to save valuable space in ROM, which could be
9B pressed after or while keying in . 27)
xlny
ulator, u ed this m
noticed by the fact that some results were not accurate to the last decimal place (e.g. 9
2). Therefore,
270
9.−can be calculated using the power function (B) as follows:
.27=\(The “Change Sign”“ key, =, must be
and also as 9270 ln.
e×− . To evaluate the latter expression press:
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hp calculators
HP 9s Powers and Roots
.27=*9H\~h
1)
swer:
(or UM.27=*9HN~h
An
6:
Both methods return 0.552528294
Example of 253 people has the same birthday as you.Find the probability that at least one
Solution: The probability that the birthday of at least one of npersons is a particular day is given by:
n
⎟
⎠
⎜
⎝
−365
1. In this example n = 253:
⎞⎛ 354
1-M NB253\
nswer:
364/365
Athat for n ≤252, the probability is
e probability of at least one of
Example 7:
0.500477154. The probability is greater than 50%. You can easily check
0%, so 253 is the smallest number of persons such that thsmaller than 5
them having the same birthday as you is greater than 50%!
ss
e element for the Mohr circle input (τxy) is
Solution:
Find the maximum shear stress on an element if the stress in the x –direction (Sx) is 25,000 psi, the stre
in the y-direction (Sy) is –5,000 psi and the shear stress on th
4,000 psi.
The maximum shear stress is given by the following formula:
2
2
yx SS ⎞
⎛−
2xy
max τ+
⎟
⎟
⎠
⎜
⎜
⎝
=τ
25A3-5=A3\/2\O+4AO\F
nswer:A 15,524.1747 psi.
Example 8: Find the geometric mean of the set of numbers { 2, 3.4, 3.41, 7, 11, 23 }
Solution: For a set of n positive numbers { a1, a2, ..., an} the geometric mean is defined as
1
n
n)a...aa(G 21 ⋅⋅⋅=
To find G press:
*3.41*7*11*23\BM1/6\
Remember that we can write the above equation this other way:
2*3.4
1Remember that the Ukey starts a new calculation. The Mkey is only active after pressing U, J, \, or a two-
argument function (e.g. +, /, B, ~Ð, etc.)
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hp calculators
HP 9s Powers and Roots
nn)a...aa(G ⋅⋅⋅= 21
which can be calculated in fewer keystrokes:
2*3.4*3.41*7*11*23\~b6\
nswer: AG =5.873725441
here cannot be more than six pending operations on the HP 9s (e.g. 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + ( 1 + generates an
? Well, let’s see if the Hp 9s can solve the following formula, which has been taken
om the cover ofthe HP-67 Owner’s Handbook and Programming Guide, and was used as an example ofthe value of
T
error condition). Is that much limitation
fr
HP’s RPN logic system.
Example 9: In a rather overoptimistic effort to break the speed of sound, high-flying pilot Ike Daedalus cranks open the
throttle on h is surplus Hawker Siddeley Harrier aircraft. From his instruments he reads a pressure altitude
(PALT) of 25,500 feet with a calibration airspeed (CAS) of 350 knots. What is the flight mach number
soundofspeed aircraftofspeed
M=
if the following formula is applicable?
[]
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
−
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
×−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎥
⎦
⎤
⎢
⎣
⎡
+= −
−112550010875611
5661
350
2015
2860
26565
6
53
2.
.
.).(
.
.M
Solution: If the problem seems very complex and you are not sure that your HP 9s can handle it, evaluate the
problem in parts, displaying intermediate results by pressing \. In this particular case, though, the
HP 9s can evaluate this formula as it is stated from left to right, without having to press the \key at all:
5500NB5.2656=+1NB.286-1N
Answer:
M5*MMMM1+.2*M350/661.5N
ONB3.5-1N*M1-6.875A=6*2
NF
M = 0.835724535
hp calculators - 5 - HP 9s Powers and Roots - Version 1.0