Home MATHEMATICS TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO

TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO

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TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO, TOPIC 4: UNITS | MATHEMATICS FORM 1, TOPIC 5: APPROXIMATIONS | MATHEMATICS FORM 1, TOPIC 6: GEOMETRY | MATHEMATICS FORM 1

TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM TWO

Exponents
Exponents tell how many times to use a number itself in multiplication. There are different laws that guides in calculations involving exponents. In this chapter we are going to see how these laws are used.
img1 1440508237210
Indication of power, base and exponent is done as follows:
img2 1440570682870
Solution:
img3 1440570912656
To write the expanded form of the following powers:
img4 1440571524913
Solution
aimg5 1440574255627
To write each of the following in power form:
img6 1440574607011
Soln.
img7 1440574709342
The Laws of Exponents
List the laws of exponents
First law:Multiplication of positive integral exponent
img10 1440575603123
Second law: Division of positive integral exponent
img23 1440583435927
Third law: Zero exponents
img30 1440590300569
Fourth law: Negative integral exponents
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Verification of the Laws of Exponents
Verify the laws of exponents
First law: Multiplication of positive integral exponent
img8 1440576676190
Generally,
when we multiply powers having the same base, we add their exponents.
If x is any base and m and n are the exponents, therefore:
img10 1440575603123
Example 1
img11 1440577017092
Solution
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If you are to write the expression using the single exponent, for example,(63)4.The expression can be written in expanded form as:
img14 1440577420739
Generally if a and b are real numbers and n is any integer,
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Example 2
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Example 3
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Example 4
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Generally, (xm)= X(mxn)
Example 5
Rewrite the following expressions under a single exponent for those with identical exponents:
img19 1440578301131
Second law: Division of positive integral exponent
Example 6
img21 1440578867884
Example 7
img22 1440583293417
Therefore,
to divide powers of the same base we subtract their exponents (subtract
the exponent of the divisor from the exponent of the dividend). That
is,
img23 1440583435927
where x is a real number and x ≠ 0, m and n are integers. m is the exponent of the dividend and n is the exponent of the divisor.
Example 8
img24 1440583736575
solution
img25 1440583736398
Third law: Zero exponents
Example 9
img26 1440583964143
This is the same as:
img27 1440583964132
If a ≠ 0, then
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Which is the same as:
img28 1440583966054
Therefore if x is any real number not equal to zero, then X0 = 1,Note that 00is undefined (not defined).
Fourth law: Negative integral exponents
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Also;
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Example 10
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Exercise 1
1. Indicate base and exponent in each of the following expressions:
img37 1440586000984
2. Write each of the following expressions in expanded form:
img38 1440586001096
3. Write in power form each of the following numbers by choosing the smallest base:
  1. 169
  2. 81
  3. 10,000
  4. 625
a. 169 b. 81c. 10 000 d. 625
4. Write each of the following expressions using a single exponent:
img39 1440586001077
5. Simplify the following expressions:
img40 1440586001040
6. Solve the following equations:
img41 1440586001133
7. Express 64 as a power with:
  1. Base 4
  2. Base 8
  3. Base 2
Base 4 Base 8 Base 2
8. Simplify the following expressions and give your answers in either zero or negative integral exponents.
img42 1440589225603
9. Give the product in each of the following:
img43 1440589350049
10. Write the reciprocal of the following numbers:
img44 1440589449987
Laws of Exponents in Computations
Apply laws of exponents in computations
Example 11
img24 1440583736575
Solution
img25 1440583736398
Re-arranging Letters so that One Letter is the Subject of the Formula
Re-arrange letters so that one letter is the subject of the formula
A formula is a rule which is used to calculate one quantity when other quantities are given. Examples of formulas are:
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22 1440666223560
Example 14
From the following formulas, make the indicated symbol a subject of the formula:
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Solution
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Transposing a Formulae with Square Roots and Square
Transpose a formulae with square roots and square
Make the indicated symbol a subject of the formula:
25 1440666704238
26 1440666843496
Exercise 3
1. Change the following formulas by making the given letter as the subject of the formula.
27 1440667111195
2. Use mathematical tables to find square root of each of the following:
28 1440667263975

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