MATHEMATICS FORM 2 TOPIC 1: EXPONENTS AND RADICALS

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MATHEMATICS FORM 2 TOPIC 1: EXPONENTS AND RADICALS

EXPONENTS AND RADICALS

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EXPONENTS AND RADICALS

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.
EXPONENTS AND RADICALS
Indication of power, base and exponent is done as follows:
EXPONENTS AND RADICALS
Solution:
EXPONENTS AND RADICALS
To write the expanded form of the following powers:
EXPONENTS AND RADICALS
Solution
EXPONENTS AND RADICALS
To write each of the following in power form:
EXPONENTS AND RADICALS
Soln.
EXPONENTS AND RADICALS
The Laws of Exponents
List the laws of exponents
First law:Multiplication of positive integral exponent
Second law: Division of positive integral exponent
Third law: Zero exponents
Fourth law: Negative integral exponents
Verification of the Laws of Exponents
Verify the laws of exponents
First law: Multiplication of positive integral exponent
EXPONENTS AND RADICALS
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:
Example 1
EXPONENTS AND RADICALS
Solution
EXPONENTS AND RADICALS
If you are to write the expression using the single exponent, for example,(63)4.The expression can be written in expanded form as:
EXPONENTS AND RADICALS
Generally if a and b are real numbers and n is any integer,
Example 2
EXPONENTS AND RADICALS
Example 3
Example 4
EXPONENTS AND RADICALS
Generally, (xm)n = X(mxn)
Example 5
Rewrite the following expressions under a single exponent for those with identical exponents:
EXPONENTS AND RADICALS
Second law: Division of positive integral exponent
Example 6
EXPONENTS AND RADICALS
Example 7
EXPONENTS AND RADICALS
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,
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
EXPONENTS AND RADICALS
solution
EXPONENTS AND RADICALS
Third law: Zero exponents
Example 9
EXPONENTS AND RADICALS
This is the same as:
EXPONENTS AND RADICALS
If a ≠ 0, then
EXPONENTS AND RADICALS
Which is the same as:
EXPONENTS AND RADICALS
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
EXPONENTS AND RADICALS
Also;
EXPONENTS AND RADICALS
Example 10
EXPONENTS AND RADICALS
Exercise 1
1. Indicate base and exponent in each of the following expressions:
EXPONENTS AND RADICALS
2. Write each of the following expressions in expanded form:
EXPONENTS AND RADICALS
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:
EXPONENTS AND RADICALS
5. Simplify the following expressions:
6. Solve the following equations:
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.
9. Give the product in each of the following:
10. Write the reciprocal of the following numbers:
EXPONENTS AND RADICALS
Laws of Exponents in Computations
Apply laws of exponents in computations
Example 11
Solution
EXPONENTS AND RADICALS
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:
EXPONENTS AND RADICALS
EXPONENTS AND RADICALS
Example 14
From the following formulas, make the indicated symbol a subject of the formula:
EXPONENTS AND RADICALS
Solution
EXPONENTS AND RADICALS
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:
EXPONENTS AND RADICALS
Exercise 3
1. Change the following formulas by making the given letter as the subject of the formula.
EXPONENTS AND RADICALS
2. Use mathematical tables to find square root of each of the following:
EXPONENTS AND RADICALS

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