CHAPTER 01 : EXPONENTS AND RADICALS

By End Of This Topic, You should be able to:

• List and Verify The laws of exponents.
• Apply laws of exponents in computations
• Simplify radicals and Perform basic operations on radicals
• Rationalize the denominator
• Read square roots and cube roots of numbers from mathematical tables
• Make one letter the subject of the formula and transpose a formula with square roots and powers

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: 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

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, (x^m)ⁿ = X^(mxn)

Example 5

Rewrite the following expressions under a single exponent for those with identical exponents:

Second law: 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,

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

solution

Third law: Zero exponents

Example 9

This is the same as:

If a ≠ 0, then

Which is the same as:

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

Also;

Example 10

Exercise 1

1. Indicate base and exponent in each of the following expressions:

2. Write each of the following expressions in expanded form:

3. Write in power form each of the following numbers by choosing the smallest base:

a.	169
b.	81
c.	10,000
d.	625

a. 169 b. 81c. 10 000 d. 625

4. Write each of the following expressions using a single exponent:

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:

Laws of Exponents in Computations

Apply laws of exponents in computations

Example 11

Solution

Radicals

Simplify radicals

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 = 5× 5;so, 5 is a square root of 25: 64 = 8× 8; 8 is a square root of

64: 216 = 6 ×6 ×6, 6 is a cube root of 216: 81 = 3 × 3 ×3 ×3,3 is a fourth root of 81: 1024 = 4 ×4 ×4 ×4 ×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 isⁿ√ where√is 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 as√without the index 2.ⁿ√pis expressed in power form as,

nth root of a number by prime factorization

Example 1, simplify the following radicals

Solution

Basic Operations on Radicals

Perform basic operations on radicals

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

If you are to find a square root of a number by using Mathematical table, first estimate the square root by grouping method. We group a given number into groups of two numbers from left. For example; to find a square root of 196 from the table, first we have to group the digits in twos from left i.e. 1 96. Then estimate the square root of the number in a group on the extreme left.

In our example it is 1. The square root of 1 is 1. Because we have two groups, this means that the answer has two digits before the decimal point. Our number is 196, read 1.9 in the table on the extreme left. From our number, we are remaining with 6, now look at the column labeled 6. Read the number where the row of 1.9 meet the column labeled 6. It meets at 1.400. Therefore the square root of 196 = 14.00 since we said that the answer must have 2 digits before decimal points. Note: If you are given a number with digits more than 4 digits. First, round off the number to four significant figures and then group the digits in twos from left. For example; the number 75678 has five digits. When we round it off into 4 digits we get 75680 and then grouping into two digits we get 7 56 80. This shows that our answer has 3 digits. We start by estimating the square root of the number to the extreme left, which is 7, the square root of 7 is between 2 and 3. Using the table, along the row 7.5, look at the column labeled 6. Read the answer where the row 7.5 meets the column 6. Then go to where it is written mean difference and look at the column labeled 8, read the answer where it meets the row 7.5. Take the first answer you got where the row 7.5 met the column 6 and add with the second answer you got where the row 7.5 met the column 8 (mean difference column). The answer you get is the square root of 75680.

Which is 275.1

Transposition of Formula

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:

Example 14

From the following formulas, make the indicated symbol a subject of the formula:

Solution

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:

Exercise 3

1. Change the following formulas by making the given letter as the subject of the formula.

2. Use mathematical tables to find square root of each of the following:

REVISION QUESTIONS

1. Simplify

Ans.

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2. By rationalizing the denominator, simplify the following expression.

Ans.

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3. Find the value of x for which 2^x ? 16 = ₈¹_x

Ans.

2^x.16=

X=-1

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4. Rationalize

Ans

Rationalize

Rationalizing the denominator we get

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5. By using mathematical tables, evaluate to three significant figures.

Ans.

Let x =

Apply log sides

Log x=

=log

=

By using Four figures Mathematical tables

Number	Log
Numerator	3.1060
	2.1822
8.39 x 10°	0.9238

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6. Simplify by rationalizing the denominator.

Ans

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7. Solve for x if

Ans.

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8. (i) Express (?3 + 5)² in the form a + b?3, where a and b are integers.

(ii) Express

in the form p + q?3, where p and q are rational numbers.

Ans

(i)

(ii) From part (i) above

Rationalizing the denominator,

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9. Solve the value of x if

Ans.

1marks

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10. Find the value of x and y if

= 2025

Ans.

= 2025

But 2025=3x3x3x3x5x5

=

So = =

Then x=2 and y=3

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11. Find the value m + n, given that 7^m × 5ⁿ = 875 .

Soln.

Given, 7^m x 5ⁿ = 875 but 875 = 7 x 125 7^m x 5ⁿ = 7 x 125

125= 5 x 5 x 5 =5³

7^m x 5ⁿ =7¹ x 5³

Now compare exponents with the same base

Hence, m=1 and n=3 m + n= 1 + 3 =4 Therefore, (m+ n) =4

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12 Which numbers are equal to their squares?

Soln.

Numbers are equal to their squares are 0 and 1.

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13. Simplify the expression

Soln.

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14. Rationalize the denominator of the expression

Ans.

Given

To rationalize the denominator, multiply by both in numerator and denominator

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15. Express in decimal notation percent of

Ans.

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16. Express R as the subject of

Ans.

Given

V (r +R) =12R

Vr+YR=12R

Vr+YR=12R

Vr=12R – VR

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17. If determine the value of

Ans

8. Given a²=12 a⁴=? a⁴=(a²)²

But a² =12

Thus a⁴ = (12)²

a⁴=144

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18. Simplify (-4g⁵)³

Soln.

(-4g⁵)³ = (-4)³ g⁸

(-4g⁵)³ = -64g⁸

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19. Rationalize the denominator to the simplest form:

Soln.

To rationalize the denominator, multiply by both in numerator and denominator

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20. Express the equation = 1 in terms of P given that P = 3^y.

Soln.

To write

in term of P given that

But

Multiply by 3 both sides

3x

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21. Find the value of x in the equation

Soln.

Given

Since bases are the same, compare exponent

Square both sides

X = 16x²

x-16x² =0

x(1 - 16x)=0

x= 0 or 1-16x=0

x=0 or x=1/16

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22. Solve for x in the equation 4^-2x × 8² = 4 × 16^x .

Soln.

4^-2x x 8² =4 x 16^x

2-4x x 26 =22 x 24x

2-4x+6 = 22+4x

Since bases are the same, now compare the exponents

-4x+6=2+4x

4x+4x=6 - 2

8x =4, divide both sides by 8

Therefore, x=0.5

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23. Solve for x in the equation 9^(x-3) × 81^(1-x) = 27^-x .

Soln.

Given Eqn. 9^(x-3) × 81^(1-x) = 27-^x .

3^2(x-3) × 3 ^4(1-x) = 3^-3x .

3^{(2x-6 +4 - 4x)} = 3^-3x

3^{(-2x - 2)} = 3^-3x

Since base are the same, now compare the exponents

-2x-2=-3x

3x-2x=2

x=2

Therefore, x=2.

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24. Simplify the expressions:

(i)

Soln.

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25. Find the values of x and y if

Soln.

2025=81 x 25= 3⁴ x 5²

From Equation above, compare exponents of the same base

-(2y-5)=2

-2y+5=2

-2y=-3

y=3/2

x+2=4

x=4-2=2

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26. Solve equation

Soln.

Since base are the same, now compare the exponents

3x= -4

x =-4/ 3= -4/3

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27. If , find the values of a, b and c.

Soln.

To get the value of a, b and c we have to rationalize the denominator

From comparison above, a= -3, b=2 and c=3

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28. Rationalize the denominator of the expression

Soln.

To rationalize the denominator, we've to multiply the given expression by in both numerator and denominator

Therefore

,

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29. If

Find values of x and y

Soln.

Given

From the above, compare exponents with the same base

x+3=-5 x=-5 -3 =-8 Therefore, x =-8

2 -y = -1

-y = -1 - 2 =-3

-y = -3, y=3

Therefore, y =3

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30. Solve for n

Soln.

Given

Since base are the same, now compare exponents

4n= 1 - n

5n =1

n = 1/5

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31. Use laws of exponents to simplify

Ans.

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32. Rationalize the denominator of

Soln

Given

To rationalize the denominator, multiply by in both numerator and denominator

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33. Find the value of x in the equation 9x3^4x =27^(x-1)

Soln.

Given 9x3^4x =27^(x-1)

3² x 3^4x = 3^3(x-1)

From laws of Exponent; a^x x a^y=a^(x+y)

3(2+4x) =3^3(x-1)

Since bases are the same, now compare the exponents

(2+4x) =3(x-1)

2+4x =3x - 3

4x - 3x=-3 -2

x= -5

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34. Without using mathematical tables, evaluate: .

Soln.

Given:

Recall Concept Of Difference of Two Square That: a ² - b² =(a +b)x (a - b)

= (0.136 + 0.148)

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35. Find the value of given that 3^a x 5^b = 675

Soln.

Given 3^a x 5^b = 675 but 625 = 25 x 27

625 =5² x3³

Now,

3^a x 5^b =3³ x 5²

Now compare

3^a = 3³ , therefore a=3

5^b = 5², therefore b= 2

Hence

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36. Solve for n in the equation =1/2

Soln.

Solve for n

By comparing exponents since base are the same

4(3-n) +(1+n)= -1

12- 4n +1+n =-1

13- 3n =-1

13+1 = 3n

14 = 3n

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37.

Soln.

To find x, take 9 to the R.H.S

since base are the same, now compare the exponents

(2x-3)=6

2x=6+3

2x=9

Therefore:

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