time ratio within the proportion and carry out its solution.
2:12 :: x:252
d. By using the rule of the extremes' product divided by the known mean to determine the
unknown mean, you will find that 12 survey teams can accomplish the task in 42 days--well before day
60.
PART B - ROOTS AND POWERS
2-3. Defining Roots and Powers. If you were asked to multiply 842 by itself and 369 by itself, it
would mean that both of these numbers were raised to their second powers. Actually, a number can be
raised to any power, and this fact is indicated by an exponent. The exponent is a number placed
somewhat above and to the right of the number. Thus, 842 x 842 can be written in its proper form of
8422. Other powers of a number are indicated by the appropriate exponent. For example, 5 x 5 x 5 x 5 x
5, which is written 55, means that the number 5 must be raised to its fifth power. Thus, the power
indicates a special case of multiplication and enables you to express numbers in a form that is useful for
special application.
a. If the power is a low-value integer, it can be readily computed by arithmetic methods.
Powers that are large numbers or decimals are usually determined through the use of logarithm tables or,
if available, computers. Low-value exponents are frequently in many of the field-survey computations,
particularly where known values are substituted for lettered terms that often must be squared (raised to
the second power). One of the formulas that a surveyor frequently uses while in the field is a2 + b2 = c2;
each letter represents a side of a right triangle (one that contains a 90€ angle). Generally, a surveyor can
lay out a right triangle and measure two sides where some obstacle or obstruction prevents measuring
the third side. Thus, if sides a and b are known, it is simple to find the value of c.
b. Equal factors of a number are known as roots; therefore, finding the root of a number is the
reverse of finding the power. When two equal factors are found for a number, each is known as the
second root, or square root. For example, the square root of 25 is 5, since 5 x 5 or 52 = 25. When three
equal factors are found, each factor is known as a third root, or cube root. The cube root of 64 is 4, since
4 x 4 x 4 or 43 = 64. Four equal factors are known as a fourth root, five equal factors are known as a
fifth root, and so on.. Square and cube roots can readily be determined by an arithmetic method.
However, only square roots are normally found by this method, since cube-root determinations are very
involved and time consuming.
c. In determining the root of a number, you must use these two notations; the radical sign and
the exponent. The radical sign ( ) is combined with the vinculum (___) and placed in front of a number
to indicate that an extraction of its square root is necessary. When a root other than a square root needs
to be determined, this is indicated by a small index number placed in the angle formed by the radical
sign as shown below.
EN0591
2-6
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