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Laws Of Indices

Class 9th Mathematics West Bengal Board Solution
Let Us Work Out 2
  1. (5 root 8)^5/2 x (16)^-3/2 Let us find out the values:
  2. (125)^-2 x (16)^-3/2^-1/6 Let us find out the values:
  3. 4^1/3 x [2^1/3 x 3^1/2] / 9^1/4 Let us find out the values:
  4. (8a^3 / 27x^-3)^2/3 x (64a^3 / 27x^-3)^- 2/3 Let us simplify:
  5. (x^-5)^2/3 - 3/10 Let us simplify:
  6. [{2 -1)-1}-1]-1 Let us simplify:
  7. cube root a^-2 b x cube root b^-2 c x cube root c^-2 a Let us simplify:…
  8. (4^m + 1/4 x root 2.2^m/2 root 2^-m)^1/m Let us simplify:
  9. 9^-3 x 16^1/4/6^-2 x (1/27)^- 4/3 Let us simplify:
  10. (x^a/x^b)^a^2 + ab+b^2 x (x^b/x^c)^b^2 + bc+c^2 x (x^c/x^a)^c^2 + ca+a^2 Let us…
  11. 5^1/2 , 10^1/4 , 6^1/3 Let us arrange in ascending order.
  12. 3^1/3 , 2^1/2 , 8^1/4 Let us arrange in ascending order.
  13. 2^60 , 3^48 , 4^36 , 5^24 Let us arrange in ascending order.
  14. (a^q/a^r)^p x (a^r/a^p)^q x (a^p/a^q)^r = 1 Let us prove
  15. (x^m/x^n)^m+n (x^n/x^exponent)^n+1 (x^exponent /x^m)^1+m = 1 Let us prove…
  16. Let us prove
  17. (a^1/x-y)^1/x-2 x (a^1/y-z)^1/y-x x (a^1/z-x)^1/z-y = 1 Let us prove…
  18. If x + z = 2y and b^2 = ac, then let us show that ay-z bz-x cx-y = 1.…
  19. If a = xyp-1, b = xyq-1 and c = xyr-1, then let us show that aq-r br-p cp-q = 1.…
  20. If x^1/a = y^1/b = z^1/c and xyz = 1, then let us show that a + b + c = 0.…
  21. If ax = by = cz and abc = 1, then let us show that xy + yz + zx = 0.…
  22. 49x = 7^3 Let us solve:
  23. 2x + 2 + 2x-1 = 9 Let us solve:
  24. 2x + 1 + 2x + 2 = 48. Let us solve:
  25. 2^4x 4^3x-1 = 4^2x/2^3x Let us solve:
  26. 9.81^x = 27^2-x Let us solve:
  27. 2^5x+4 - 2^9 = 2^10 Let us solve:
  28. 62x + 4 = 33x . 2x + 8 Let us solve:
  29. The value of (0.243)0.2 × (10)0.6 isA. 0.3 B. 3 C. 0.9 D. 9
  30. The value of isA. 1 B. 2 C. 4 D. 1/2
  31. If 4x = 8^3 , then the value of x isA. 3/2 B. 9/2 C. 3 D. 9
  32. If 20^-x = 1/7 then the value of (20)2x isA. 1/49 B. 7 C. 49 D. 1…
  33. If 4 × 5x = 500, then the value of xx isA. 8 B. 1 C. 64 D. 27
  34. If (27)x = (81)y, then let us write x : y.
  35. If (5^5 + 0.01)^2 - (5^5 -0.01)^2 = 5x, then let us calculate the value of x and…
  36. If 3.27x = 9x + 4, then let us calculate the value of x and write it.…
  37. Let us find out the value of 3 root (1/64)^1/2 and write it.
  38. Let us write explaining the greater value between 3^3^3 and (3^3)^3 with reason.…
Let Us Work Out 2
Question 1.

Let us find out the values:
Answer:











Question 2.

Let us find out the values:
Answer:


=


=


=


= = 10
Question 3.

Let us find out the values:
Answer:








Question 4.

Let us simplify:
Answer:










Question 5.

Let us simplify:
Answer:






Question 6.

Let us simplify:

[{2 –1)–1}–1]–1
Answer:

[{2 –1)–1}–1]–1




= 2
Question 7.

Let us simplify:
Answer:








Question 8.

Let us simplify:
Answer:









Question 9.

Let us simplify:
Answer:








Question 10.

Let us simplify:
Answer:








Question 11.

Let us arrange in ascending order.
Answer:

In this type of problems we try to make the powers same by taking LCM of them.

Here, are in the powers .


So, we take LCM of 2, 4 and 3.


4= 22


2= 21


3= 31


LCM= 23 = 6


Now,




Here, all the numbers are having same power.


So, in ascending order we have



Hence, .
Question 12.

Let us arrange in ascending order.
Answer:

In this type of problems we try to make the powers same by taking LCM of them.

Here, are in the powers .


So, we take LCM of 2, 4 and 3.


4= 22


2= 21


3= 31


LCM= 23 = 6


Now,




Here, all the numbers are having same power.


So, in ascending order we have



Hence, .
Question 13.

Let us arrange in ascending order.

260, 348, 436, 524
Answer:

We will try to make the powers of the same . We can write

260 = = 3212


348 = = 8112


436 = = 6412


324 = = 912


Now, we can clearly see


So, .
Question 14.

Let us prove

= 1
Answer:

LHS =





Therefore, LHS = RHS


Hence, proved.
Question 15.

Let us prove

= 1
Answer:

LHS =





Therefore, LHS = RHS


Hence, proved .
Question 16.

Let us prove
Answer:

LHS =






=(x0) = 1 = RHS


Therefore, LHS = RHS


Hence, proved .
Question 17.

Let us prove
Answer:












Therefore, LHS = RHS


Hence, proved.
Question 18.

If x + z = 2y and b2 = ac, then let us show that ay–z bz–x cx–y = 1.
Answer:

Given that x + z = 2y and b2 = ac

We can write from x + z = 2y


…eq(1)


Now, b2 = ac





Now, we can write as


and …eq(2)


Now, LHS = ay–z bz–x cx–y






= b0 = 1 = RHS


Therefore, LHS = RHS


Hence, proved .
Question 19.

If a = xyp–1, b = xyq–1 and c = xyr–1, then let us show that aq–r br–p cp–q = 1.
Answer:

Given, a = xyp–1, b = xyq–1 and c = xyr–1


Now, LHS = aq–r br–p cp–q


=


= )


=


=


= x0 × y0 = 1 × 1 = 1 = RHS


Therefore, LHS = RHS


Hence, proved.
Question 20.

If and xyz = 1, then let us show that a + b + c = 0.
Answer:

Given and xyz = 1

Let


Therefore,


…eq(1)



…eq(2)



…eq(3)


Now, xyz = 1



(from eq(1), eq(2)and eq(3))




We know that if base of the exponents are same then the powers are also equal.


Therefore, a + b + c = 0.


Hence the relation is proved.
Question 21.

If ax = by = cz and abc = 1, then let us show that xy + yz + zx = 0.
Answer:

Given that : -


ax = by = cz = k and abc = 1


Formula used: - (a1) if then


(b1)


………(i)


……… (ii)


……… (iii)


multiplying (i),(ii)&(iii) we get


using formula (a1)



using formula (b1)





…………..proved
Question 22.

Let us solve:

49x = 73
Answer:

Given that 49x = 73


Formula used: - (a1) if then a = b


Solving it



We can write


Then ,


Using the formula (a1)
Question 23.

Let us solve:

2x + 2 + 2x–1 = 9
Answer:

Given that 2x + 2 + 2x–1 = 9


Formula used: - (a1) if then a = b


2x + 2 + 2x–1 = 23 + 1



Using formula (a1)


and


x = 1
Question 24.

Let us solve:

2x + 1 + 2x + 2 = 48.
Answer:

Given that



Formula used: - (a1) if then a = b



and




Question 25.

Let us solve:
Answer:

Given that



Formula used: - (a1) if then a = b


(b1)
Question 26.

Let us solve:
Answer:

Given that


Formula used: - (a1) if then


(b1)
Question 27.

Let us solve:
Answer:

GIVEN THAT


Formula used: - (a1) if then


(b1)




⇒ 5x + 4 = 9


⇒ x = 1
Question 28.

Let us solve:

62x + 4 = 33x . 2x + 8
Answer:

Given that 62x + 4 = 33x . 2x + 8


Formula used: - (a1) if then



and




Question 29.

The value of (0.243)0.2 × (10)0.6 is
A. 0.3

B. 3

C. 0.9

D. 9
Answer:

Formula used: - (a1) if then


(b1)


(0.243)0.2 × (10)0.6
Question 30.

The value of is
A. 1

B. 2

C. 4

D.


Answer:

Formula used: - (a1) if then


(b1)






= 22 = 4


Hence, option (c) is correct.
Question 31.

If 4x = 83, then the value of x is
A.

B.

C. 3

D. 9
Answer:

Formula used: - (a1) if then


(b1)


……… (i)


And ) ……(ii)


From (i)(ii)
Question 32.

If then the value of (20)2x is
A.

B. 7

C. 49

D. 1
Answer:

Formula used: - (a1) if then


(b1)





So,


Question 33.

If 4 × 5x = 500, then the value of xx is
A. 8

B. 1

C. 64

D. 27
Answer:

Formula used: - (a1) if then


(b1)






According to question
Question 34.

If (27)x = (81)y, then let us write x : y.
Answer:

Formula used: - (a1) if then


(b1)


(27)x = 33x …… (i)


(81)y = 34y …… (ii)


From (i)&(ii) we get


33x = 34y


3x = 4y



According to question


x:y = 4:3
Question 35.

If (55 + 0.01)2 – (55 –0.01)2 = 5x, then let us calculate the value of x and write it.
Answer:

Formula used: - (a1) if then


(b1)


(c1)


(d1)



Using formula (c1)&(d1)



Using formula (b1)



from formula (a1)
Question 36.

If 3.27x = 9x + 4, then let us calculate the value of x and write it.
Answer:

Formula used: - (a1) if then





Using formula (a1)
Question 37.

Let us find out the value of and write it.
Answer:

Formula used: - (a1) if then


(b1)


Solving the equation






=



Question 38.

Let us write explaining the greater value between and with reason.
Answer:

Given equation are and


Let as find out the value of



…… (a)


But


…… (b)



Now if we compare with then



From (a) and (b)



Therefore are from comparing the exact value.