Home MATHEMATICS TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM 2

# TOPIC 1: EXPONENTS AND RADICALS ~ MATHEMATICS FORM 2

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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.

Indication of power, base and exponent is done as follows:
Solution:
To write the expanded form of the following powers:
Solution
To write each of the following in power form:
Soln.
The Laws of Exponents
List the laws of exponents
First law:Multiplication of positive integral exponent
Second law:Â Division of positive integral exponent
Third law:Â Zero exponents
Fourth law:Â Negative integral exponents
Verification of the Laws of Exponents
Verify the laws of exponents
First law:Â Multiplication of positive integral exponent
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
Solution
If you are to write the expression using the single exponent, for example,(63)4.The expression can be written in expanded form as:
Generally if a and b are real numbers and n is any integer,
Example 2
Example 3
Example 4
Generally, (xm)nÂ = X(mxn)
Example 5
Rewrite the following expressions under a single exponent for those with identical exponents:
Second law:Â Division of positive integral exponent
Example 6
Example 7
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,
whereÂ xÂ is a real number andÂ xÂ â‰  0, m and n are integers. m is the exponent of the dividend and n is the exponent of the divisor.
Example 8
solution
Third law:Â Zero exponents
Example 9
This is the same as:
If a â‰  0, then
Which is the same as:
Therefore ifÂ xÂ is any real number not equal to zero, then X0Â = 1,Note that 00is undefined (not defined).
Fourth law:Â Negative integral exponents
Also;
Example 10
Exercise 1
1.Â Indicate base and exponent in each of the following expressions:
2.Â Write each of the following expressions in expanded form:
3.Â 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.Â Write each of the following expressions using a single exponent:
5.Â Simplify the following expressions:
6.Â Solve the following equations:
7.Â Express 64 as a power with:
1. Base 4
2. Base 8
3. Base 2
Base 4 Base 8 Base 2
8.Â Simplify the following expressions and give your answers in either zero or negative integral exponents.
9.Â Give the product in each of the following:
10.Â Write the reciprocal of the following numbers:
Laws of Exponents in Computations
Apply laws of exponents in computations
Example 11
Solution
Radicals are opposite of exponents. For example when we raise 2 by 2 we get 4 but taking square root of 4 we get 2. The same way we can raise the number using any number is the same way we can have the root of that number. For example, square root, Cube root, fourth root, fifth roots and so on. We can simplify radicals if the number has factor with root, but if the number has factors with no root then it is in its simplest form. In this chapter we are going to learn how to find the roots of the numbers and how to simplify radicals.
When a number is expressed as a product of equal factors, each of the factors is called the root of that number. For example,25 = 5Ã— 5;so, 5 is a square root of 25: 64 = 8Ã— 8; 8 is a square root of 64: 216 = 6 Ã—6 Ã—6, 6 is a cube root of 216: 81 = 3 Ã— 3 Ã—3 Ã—3,3 is a fourth root of 81: 1024 = 4 Ã—4 Ã—4 Ã—4 Ã—4, 4 is a fifth root of 1024.
Therefore, the nth root of a number is one of the n equal factors of that number. The symbol for nth root isnâˆš whereâˆšis called a radical and n is the index (indicates the root you have to find). If the index is 2, the symbol represents square root of a number and it is simply written asâˆšwithout the index 2.nâˆšpis expressed in power form as,
nth root of a number by prime factorization
Example 1, simplify the following radicals
Soution
Different operations like addition, multiplication and division can be done on alike radicals as is done with algebraic terms.
Simplify the following:
Solution
Simplify each of the following expressions:
group the factors into groups of two equal factors and from each group take one of the factors.
The Denominator
Rationalize the denominator
If you are given a fraction expression with radical value in the denominator and then you express the expression given in such a way that there are no radical values in the denominator, the process is called rationalization of the denominator.
Example 12
Rationalize the denominator of the following expressions:
Example 13
Rationalize the denominator for each of the following expressions:
Solution
To rationalize these fractions, we have to multiply by the fraction that is equals to 1. The factor should be considered by referring the difference of two squares.
Exercise 2
1. Simplify each of the following by making the number inside the radical sign as small as possible:
Square Roots and Cube Roots of Numbers from Mathematical Tables
Read square roots and cube roots of numbers from mathematical tables