Indices And Cube Root Class 8th Mathematics (new) MHB Solution

Indices And Cube Root

Class 8th Mathematics (new) MHB Solution
Practice Set 3.1


Practice Set 3.1

Question 1.

Express the following numbers in index form.

(1) Fifth root of 13

(2) Sixth root of 9

(3) Square root of 256

(4) Cube root of 17

(5) Eighth root of 100

(6) Seventh root of 30


Answer:

(1) Fifth root of 13


In general, nth root of ‘a’ is expressed as .


So, the fifth root of 13 is expressed as .


Here, 13 is base,  is the index and  is the index form of the number.


(2) Sixth root of 9


In general, nth root of ‘a’ is expressed as .


So, the sixth root of 9 is expressed as .


Here, 9 is base,  is the index and  is the index form of the number.


(3) Square root of 256


In general, nth root of ‘a’ is expressed as .


So, the square root of 256 is expressed as .


Here, 256 is base,  is the index and  is the index form of the number.


(4) Cube root of 17


In general, nth root of ‘a’ is expressed as .


So, cube root of 17 is expressed as.


Here, 17 is base, is the index and  is the index form of the number.


(5) Eighth root of 100


In general, nth root of ‘a’ is expressed as .


So, the eighth root of 100 is expressed as .


Here, 100 is base,  is the index and  is the index form of the number.


(6) Seventh root of 30


In general, nth root of ‘a’ is expressed as .


So, the seventh root of 30 is expressed as .


Here, 30 is base,  is the index and  is the index form of the number.



Question 2.

Write in the form ‘nth root of a’ in each of the following numbers.

1. (81)1/4 2. (49)1/2

3. (15)1/5 4. (512)1/9

5. (100)1/19 6. (6)1/7


Answer:

1. (81)1/4


In general, a1/n is written as ‘nth root of a’.


So, (81)1/4 is written as ‘4th root of 81’.


2. (49)1/2


In general, a1/n is written as ‘nth root of a’.


So, (49)1/2 is written as ‘square root of 49’.


3. (15)1/5


In general, a1/n is written as ‘nth root of a’.


So, (15)1/5 is written as ‘5th root of 15’.


4. (512)1/9


In general, a1/n is written as ‘nth root of a’.


So, (512)1/9 is written as ‘9th root of 512’.


5. (100)1/19


In general, a1/n is written as ‘nth root of a’.


So, (100)1/19 is written as ‘19th root of 100’.


6. (6)1/7


In general, a1/n is written as ‘nth root of a’.


So, (6)1/7 is written as ‘7th root of 6’.




Practice Set 3.2

Question 1.

Complete the following table.



Answer:


Explanation of Table


Generally we can express two meaning of the number am/n.


am/n = (am)1/n means ‘nth root of mth powerof a’.


am/n = ()m means ‘mth power of nth root of a’.


(1) (225)3/2


(2253)1/2 means ‘Cube of square root of 225’.


(2251/2)3 means ‘Square root of cube of 225’.


(2) (45)4/5


(454)1/5 means ‘Fourth power of fifth root of 45’.


(451/5)4 means ‘Fifth root of fourth power of 45’.


(3) (81)6/7


(816)1/7 means ‘Sixth power of seventh root of 81’.


(811/7)6 means ‘Seventh root of sixth power of 81’.


(4) (100)4/10


(1004)1/10 means ‘Fourth power of tenth root of 100’.


(1001/10)4 means ‘Tenth root of fourth power of 100’.


(5) (21)3/7


(213)1/7 means ‘Cube of seventh root of 21’.


()3 means ‘Seventh root of cube of 21’.



Question 2.

Write the following number in the form of rational indices.

(1) Square root of 5th power of 121.

(2) Cube of 4th root of 324.

(3) 5th root of square of 264.

(4) Cube of cube root of 3.


Answer:

we know that ‘nth root of mth powerof a’ is expressed as (am)1/n


and ‘mth power of nth root of a’ is expressed as ()m.


(1) Square root of 5th power of 121.


We know that,


‘nth root of mth powerof a’ is expressed as (am)1/n


So, ‘Square root of 5th power of 121’ is expressed as (1215)1/2 or (121)5/2.


(2) Cube of 4th root of 324.


We know that,


‘nth root of mth powerof a’ is expressed as (am)1/n


So, ‘Cube of 4th root of 324’ is written as (3241/4)3 or (324)3/4.


(3) 5th root of square of 264.


We know that,


‘nth root of mth powerof a’ is expressed as (am)1/n


So, ‘5th root of square of 264’ is written as (2642)1/5 or


(264)2/5.


(4) Cube of cube root of 3.


We know that,


‘mth power of nth root of a’ is expressed as ()m


So, ‘Cube of cube root of 3’ is written as (31/3)3 or (31)3/3.




Practice Set 3.3

Question 1.

Find the cube root of the following numbers.

8000


Answer:

First find the factor of 8000


8000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5


For finding the cube root, we pair the prime factors in 3’s.


= (2 × 2 × 5)3


= (2 × 10)3


= 203


i.e. cube root of 8000 = (8000)1/3 = (203)1/3 = 20 (answer).



Question 2.

Find the cube root of the following numbers.

729


Answer:

First find factors of 729


729 = 9 × 9 × 9


For finding the cube root, we pair the prime factors in 3’s.


= 93


i.e. cube root of 729 = (729)1/3 = (93)1/3 = 9 (answer).



Question 3.

Find the cube root of the following numbers.

343


Answer:

First find the factor of 343


343 = 7 × 7 × 7


For finding the cube root, we pair the prime factors in 3’s.


= 73


i.e. cube root of 343 = (343)1/3 = (73)1/3 = 7 (answer).



Question 4.

Find the cube root of the following numbers.

-512


Answer:

First find factors of - 512


-512 = (-8) × (-8) × (-8)


For finding the cube root, we pair the prime factors in 3’s.


= (-8)3


i.e. cube root of -512 = (- 512)1/3 = (-83)1/3 = -8 (answer).



Question 5.

Find the cube root of the following numbers.

-2744


Answer:

First find factors of -2744


-2744 = (-14) × (-14) × (-14)


For finding the cube root, we pair the prime factors in 3’s.


= (-14)3


i.e. cube root of -2744 = (- 2744)1/3 = (-143)1/3 = -14 (answer).



Question 6.

Find the cube root of the following numbers.

32768


Answer:

First find factor of 32768


32768 = 32 × 32 × 32


For finding the cube root, we pair the prime factors in 3’s.


= 323


i.e. cube root of 32768 = ∛32768 = (323)1/3 = 32 (answer).



Question 7.

Simplify:

(1) 

(2) 

(3) If  = 9 then  = 


Answer:

(1) 


(answer).


(2) 


 (answer).


3) If ∛729 = 9 then ∛0.000729 = ?



We know that ∛729 = 9


S0 (answer).


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