Q.1. Assuming that x, y, z are positive real numbers, simplify each of the following:
(i) (√x−3)5
(ii) √x3 y−2
(iii) (x−2/3y−1/2)2
(iv) (√x)−2/3√y4 ÷ √xy−1/2
(v) 5√243 x10y5z10
(vi)
(vii)
Proof: We have to simplify the following, assuming thatare positive real numbers
(i) Given
As x is positive real number then we have
Hence the simplified value ofis
(ii) Given
By using law of rational exponentswe have
Hence the simplified value ofis
(iii) Given
As x and y are positive real numbers then we have
By using law of rational exponentswe have
By using law of rational exponentswe have
Hence the simplified value ofis
(iv)
by using the law of rational exponents, am ÷ an = am−n, we have
(v)
=(243 × x10 × y5 × z10)1/5
=(243)1/5 × (x10)1/5 × (y5)1/5 × (z10)1/5
=(35)1/5 × x10×1/5 × y5×1/5 × z10×1/5
=3 × x2 × y × z2
=3x2yz2
(vi)
(vii)
Q.2. Simplify:
(i)
(ii)
(iii)
(v)
(v)
(vi)
(vii)
Proof: (1) Given
By using law of rational exponentswe have
Hence the value ofis
(ii)
(iii) Given
Hence the value ofis
(iv) Given
The value ofis
(v) Given
(vi) GivenSo,
By using the law of rational exponents
Hence the value ofis
(vii) GivenSo,
Hence the value ofis
Q.3. Prove that:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
Proof: (i) We have to prove that
By using rational exponentwe get,
Hence,
(ii) We have to prove that
Hence,
iii) We have to prove that
Now,
Hence,
(iv) We have to prove that
Let
Hence,
(v) We have to prove that
Let
Hence,
(vi) We have to prove thatSo,
Let
Hence,
(vii) We have to prove that So let
Hence,
(viii) We have to prove thatSo,
Let
Hence,
(ix) We have to prove that
Let
Hence,
Q.4. Show that:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Proof: (i)
= 1
(ii)
= 1
(iii)
(iv)
=x2(a3+b3+c3)
(v)
= x0
= 1
(vi)
= x1
= x
(vii)
=a0
=1
(viii)
=30
=1
Q.5. If 27x =9/3x, find x
Proof: We are givenWe have to find the value of x
Since
y using the law of exponents we get,
On equating the exponents we get,
Hence,
Q.6. Find the values of x in each of the following:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii) 52x+3=1
(viii)
(ix)
Proof: From the following we have to find the value of x
(i) Given
By using rational exponents
On equating the exponents we get,
The value of x is
(ii) Given
On equating the exponents
Hence the value of x is
(iii) Given
Comparing exponents we have,
Hence the value of x is
(iv) Given
On equating the exponents of 5 and 3 we get,
And,
The value of x is
(v) Given
On equating the exponents we get
And,
Hence the value of x is
(vi)
On comparing we get,
⇒4x+1 = −15
⇒4x = −16
⇒x = −4
(vii) 52x+3=1
52x+3 =50
⇒2x+3 = 0
⇒x = −3/2
(viii) (13)√x = 44 − 34 − 6
On comparing we get,
√x = 2
on squaring both sides we get,
x = 4
(ix)
On comparing we get,
⇒x + 1 = −6
⇒x = −7
Q.7. If 34x = (81)−1 and 101/y = 0.0001, find the value of 2−x+4y.
Proof: It is given that 34x=(81)−1 and 101/y=0.0001.
Now,
34x = (81)−1
⇒34x = (34)−1
⇒(3x)4 = (3−1)4
⇒x = −1
And,
101/y=0.0001
⇒101/y = (1/10)4
⇒101/y = (10)−4
⇒1/y = −4
⇒y = −1/4
Therefore, the value of 2−x+4y is 21+4(−1/4) = 20 = 1
Q.8. If 53x=125 and 10y=0.001, find x and y.
Proof: It is given that 53x = 125 and 10y = 0.001.
Now,
53x = 125
⇒53x = 53
⇒3x = 3
⇒x = 1
And,
10y=0.001
⇒10y = 1/1000
⇒10y = 10−3
⇒y = −3
Hence, the values of x and y are 1 and −3, respectively.
Q.9. Solve the following equations:
(i) 3x+1 = 27×34
(ii)
(iii) 3x−1 × 52y−3 = 225
(iv) 8x+1 = 16y+2 and, (1/2)3+x = (1/4)3y
(v) 4x−1 × (0.5)3−2x = (178)x
(vi)where a and b are distinct primes
Proof: (i)
3x+1 = 27×34
⇒3x+1 = 33×34
⇒3x+1 = 33+4
⇒3x+1 = 37
⇒x+1 = 7
⇒x = 6
(ii)
(iii)
3x−1 × 52y−3 = 225
⇒3x−1 × 52y−3 = 3 × 3 × 5 × 5
⇒3x−1 × 52y−3 = 32 × 52
⇒x − 1 = 2 and 2y − 3 = 2
⇒ x = 3 and y = 5/2
(iv)
⇒3x+3 = 4y+8 and 3+x = 6y
Now,
3+x = 6y ⇒ x = 6y − 3
Putting x = 6y − 3 in 3x − 4y = 5, we get
3(6y−3)−4y = 5
⇒18y − 9 − 4y = 5
⇒14y = 14
⇒y = 1
Putting y = 1 in x=6y−3, we get
x=6×1−3=3
(v)
⇒4x − 5 = −3x
⇒7x = 5
⇒x = 5/7
(vi)
⇒1/2 = 2x − 1
⇒3/2 = 2x
⇒x = 3/4
Q.10. If a and b are distinct primes such that= axb2y, find x and y.
Proof: Given: = axb2y
Q.11. If a and b are different positive primes such that
(i)= axby, find x and y.
(ii)= axby, find x + y + 2.
Proof: (i)
⇒a−21−5b42+8 = axby
⇒a−26b50 = axby
⇒x = −26 and y = 50
(ii)
⇒(ab)−1 = axby
⇒a−1b−1 = axby
⇒x=−1 and y=−1
Therefore, the value of x + y + 2 is −1 −1 + 2 = 0.
Q.12. If 2x × 3y × 5z = 2160, find x , y and z. Hence, compute the value of 3x × 2− y × 5− z.
Proof: Given: 2x × 3y × 5z = 2160
First, find out the prime factorization of 2160.
It can be observed that 2160 can be written as 24 × 33 × 51.
Also,
2x × 3y × 5z = 24 × 33 × 51
⇒x = 4, y = 3, z = 1
Therefore, the value of 3x × 2−y × 5−z is 34 × 2−3 × 5−1 = 81
Q.13. If 1176 = 2a × 3b × 7c, find the values of a, b and c. Hence, compute the value of 2a × 3b × 7−c as a fraction.
Proof: First find the prime factorisation of 1176.
It can be observed that 1176 can be written as 23 × 31 × 72.
1176 = 2a3b7c = 233172
So, a = 3, b = 1 and c = 2.
Therefore, the value of 2a × 3b × 7−c is 23 × 31 × 7−2= 8 x 3 x 1/49 =
Q.14. Simplify:
(i)
(ii)
Proof: (i)
= 1
(ii)
= x0
= 1
Q.15. Show that:
Proof:
Q.16. (i) If a = xm+nyl, b = xn+lym and c = xl+myn, prove that am−nbn−lcl−m = 1.
(ii) If x = am+n, y = an+l and z = al+m, prove that xmynzl=xnylzm.
Proof: (i) Given: a = xm+nyl, b = xn+lym and c = xl+myn
= x0y0
= 1
(ii) Given:x = am+n, y = an+l and z = al+m
Putting the values of x, y and z in xmynzl, we get
xmynzl
=am2+n2+l2+nm+ln+lm
Putting the values of x, y and z in xnylzm, we get
xnylzm
So, xmynzl = xnylzm
Multiple Choice Questions(MCQs)
Q.1. The value ofis
(a) 5
(b) 125
(c) 1/5
(d) -125
Proof: We have to find the value of
So,
The value ofis 125
Hence the correct choice is
Q.2. The value of x − yx-y when x = 2 and y = −2 is
(a) 18
(b) −18
(c) 14
(d) −14
Proof: Given
Here x = 2, y = -2
By substituting in we get
The value ofis -14
Hence the correct choice is
Q.3. The product of the square root of x with the cube root of x is
(a) cube root of the square root of x
(b) sixth root of the fifth power of x
(c) fifth root of the sixth power of x
(d) sixth root of x
Proof: We have to find the product (say L) of the square root of x with the cube root of x is. So,
The product of the square root of x with the cube root of x is
Hence the correct alternative is
Q.4. The seventh root of x divided by the eighth root of x is
(a) x
(b) √x
(c)
(d)
Proof: We have to find he seventh root of x divided by the eighth root of x, so let it be L. So,
The seventh root of x divided by the eighth root of x is
Hence the correct choice is
Q.5. The square root of 64 divided by the cube root of 64 is
(a) 64
(b) 2
(c) 1/2
(d) 642/3
Proof: We have to find the value of
So,
The value ofis 2
Hence the correct choice is
Q.6. Which of the following is (are) not equal to
(a)
(b)
(c)
(d)
Proof: We have to find the value of
So,
Hence the correct choice is
Q.7. When simplified (x−1+y−1)−1 is equal to
(a) xy
(b) x+y
(c) xy/x+y
(d) x+y/xy
Proof: We have to simplify
So,
The value ofis
Hence the correct choice is
Q.8. If 8x+1 = 64 , what is the value of 32x+1 ?
(a) 1
(b) 3
(c) 9
(d) 27
Proof: We have to find the value ofprovided
So,
Equating the exponents we get
By substitute inwe get
The real value ofis 27
Hence the correct choice is
Q.9. If (23)2 = 4x, then 3x =
(a) 3
(b) 6
(c) 9
(d) 27
Proof: We have to find the value of 3x provided(23)2 = 4x
So,
23x2 = 22x
26 = 22x
By equating the exponents we get
By substituting in 3x we get
The value of 3x is 27
Hence the correct choice is
Q.10. If x-2 = 64, then x1/3+x0 =
(a) 2
(b) 3
(c) 3/2
(d) 2/3
Proof: We have to find the value of if
Consider,
Multiply on both sides of powers we get
By taking reciprocal on both sides we get,
Substitutinginwe get,
By taking least common multiply we get
Hence the correct choice is
Q.11. When simplifiedis
(a) 9
(b) −9
(c) 1/9
(d) −1/9
Proof: We have to find the value of
So,
Hence the correct choice is
Q.12. Which one of the following is not equal to?
(a)
(b)
(c)
(d)
Proof: We have to find the value of
So,
Also,
Hence the correct alternative is
Q.13. Which one of the following is not equal to?
(a)
(b)
(c)
(d)
Proof: We have to find the value of
So,
Since,is equal to ,,
Hence the correct choice is
Q.14. If a, b, c are positive real numbers, then is equal to
(a) 1
(b) abc
(c) √abc
(d) 1/abc
Proof: We have to find the value of when a, b, c are positive real numbers.
So,
Taking square root as common we get
Hence the correct alternative is
Q.15. , then x =
(a) 2
(b) 3
(c) 4
(d) 1
Proof: We have to find value of x provided
So,
Equating exponents of power we get x = 4
Hence the correct alternative is
Q.16. The value of is
(a) 1/2
(b) 2
(c) 1/4
(d) 4
Proof: Find the value of
Hence the correct choice is
Q.17. If a, b, c are positive real numbers, thenis equal to
(a) 5a2bc2
(b) 25ab2c
(c) 5a3bc3
(d) 125a2bc2
Proof: Find value of
Hence the correct choice is
Q.18. If a, m, n are positive integers, thenis equal to
(a) amn
(b) a
(c) am/n
(d) 1
Proof: Find the value of
So,
Hence the correct choice is
Q.19. If x = 2 and y = 4, then=
(a) 4
(b) 8
(c) 12
(d) 2
Proof: We have to find the value of
Substitute x = 2, y = 4 into get
Hence the correct choice is
Q.20. The value of m for which=7m, is
(a) −1/3
(b) 1/4
(c) −3
(d) 2
Proof: We have to find the value of for
By using rational exponents
7−1/3=7m
Equating power of exponents we get
Hence the correct choice is
Q.21. The value of, is
(a) 196
(b) 289
(c) 324
(d) 400
Proof: We have to find the value of
By using the identity we get,
Hence correct choice is
Q.22. (256)0.16 × (256)0.09
(a) 4
(b) 16
(c) 64
(d) 256.25
Proof: We have to find the value of
By using law of rational exponents
The value ofis 4
Hence the correct choice is
Q.23. If , then 10-y equals
(a) −1/5
(b) 1/50
(c) 1/625
(d) 1/5
Proof: We have to find the value of 10-y
Given that 102y = 25 therefore,
Hence the correct option is.
Q.24. If 9x+2 = 240 + 9x, then x = z
(a) 0.5
(b) 0.2
(c) 0.4
(d) 0.1
Proof: We have to find the value of x
Given, 9x+2 = 240 + 9x
32x = 31
By equating the exponents we get
2x = 1
x = 1/2
x = 0.5
Hence the correct alternative is.
Q.25. If x is a positive real number and x2 = 2, then x3 =
(a) √2
(b) 2√2
(c) 3√2
(d) 4
Proof: We have to find x3 provided x2 = 2. So,
By raising both sides to the power 1/2
By substitutingin x3 we get
The value of x3 is
Hence the correct choice is.
Q.26. Ifand x > 0, then x =
(a) √2/4
(b) 2√2
(c) 4
(d) 64
Proof: For, we have to find the value of x
So,
By raising both sides to the power 2 we get
The value of x is 64
Hence the correct alternative is
Q.27. If g = t2/3 + 4t − 1/2, What is the value of g when t = 64?
(a) 21/2
(b) 33/2
(c) 16
(d) 257/16
Proof: GivenWe have to find the value of
So,
The value of g is 33/2
Hence the correct choice is
Q.28. If 4x − 4x−1 = 24, then (2x)x equals
(a) 5√5
(b) √5
(c) 25√5
(d) 125
Proof: We have to find the value of
So,
Taking 4x as common factor we get
By equating powers of exponents we get
By substitutingin we get
Hence the correct choice is
Q.29. When simplifiedis
(a) 8
(b) 1/8
(c) 2
(d) 1/2
Proof:
Hence the correct choice is.
Q.30. If, then x =
(a) 2
(b) 3
(c) 5
(d) 4
Proof: We have to find the value of x provided
So,
By cross multiplication we get
By equating exponents we get
And,
Hence the correct choice is
Q.31. The value of 64-1/3 (641/3-642/3), is
(a) 1
(b) 1/3
(c) −3
(d) −2
Proof: Find the value of
So,
Hence the correct statement is.
Q.32. If √5n = 125, then =
(a) 25
(b) 1/125
(c) 625
(d) 1/5
Proof: We have to findprovided
So,
Substitute n = 6 in to get
Hence the value ofis 25
The correct choice is
Q.33. If (16)2x+3 =(64)x+3, then 42x-2 =
(a) 64
(b) 256
(c) 32
(d) 512
Proof: We have to find the value of 42x-2 provided
So,
Equating the power of exponents we get
x = 3
The value of 42x-2 is
Hence the correct alternative is
Q.34. If thenis equal to
(a) 1/2
(b) 2
(c) 4
(d) −1/4
Proof: We have to find the value ofprovided
Consider,
Equating the power of exponents we get
2m = 2
m = 2/2
m = 1
By substituting we get
Hence the correct choice is.
Q.35. If,and a = 21/10 , then=
a) 2
(b) 1/4
(c) 9
(d) 1/8
Proof: Given :, and a = 21/10
To find :
Find :
By using rational components We get
By equating rational exponents we get
Now,= (a2m+n−p).(am−2n+2p) we get
=a2m+n−p+m−2n+2p
=a3m−n+p
Now putting value of a = 21/10 we get,
On comparing LHS and RHS we get, p - n = 4.
Now,
= a3m - n + p
= 2
So, option (a) is the correct answer.
Q.36. If=37 , then x
(a) 3
(b) −3
(c) 1/3
(d) −1/3
Proof: We have to find the value of x provided
So,
By using law of rational exponents we get
By equating exponents we get
Hence the correct choice is.
Q.37. If o <y <x, which statement must be true?
(a) √x−√y = √x−y
(b) √x + √x = √2x
(c) x√y = y√x
(d) √xy = √x√y
Proof: We have to find which statement must be true?
Given 0<y<x,
Option (a) :
Left hand side:
Left hand side is not equal to right hand side
The statement is wrong.
Option (b) :
Left hand side:
Right Hand side:
Left hand side is not equal to right hand side
The statement is wrong.
Option (c) :
Left hand side:
Right Hand side:
Left hand side is not equal to right hand side
The statement is wrong.
Option (d) :
Left hand side:
Right Hand side:
Left hand side is equal to right hand side
The statement is true.
Hence the correct choice is
Q.38. If 10x = 64, what is the value of 10x/2+1 ?
(a) 18
(b) 42
(c) 80
(d) 81
Proof: We have to find the value ofprovided
So,
By substituting we get
Hence the correct choice is.
Q.39. is equal to
(a) 5/3
(b) −5/3
(c) 3/5
(d) −3/5
Proof: We have to simplify
Taking 5n as a common factor we get
Hence the correct alternative is.
Q.40. Ifthen=
(a) 3
(b) 9
(c) 27
(d) 81
Proof: We have to find
Given
Equating powers of rational exponents we get
Substituting inwe get
Hence the correct choice is.
Fill in the Blanks Types of Questions(FBQs)
Q.1. (212 – 152)2/3 is equal to ________.
Proof:
= (6)2
= 36
Hence, (212 – 152)2/3 is equal to 36.
Q.2. 811/4 × 93/2 × 27–4/3 is equal to _________.
Proof:
= 1
Hence, 811/4 × 93/2 × 27–4/3 is equal to 1.
Q.3. = __________.
Proof:
Hence,
Q.4. If x = 82/3 × 32–2/5, then x–5 = ________.
Proof: Let x =
⇒x =
⇒x =
⇒x =
⇒x =
⇒x = 1
Now,
= 1
Hence, if x = 82/3 × 32–2/5, then x–5 = 1.
Q.5. If 6n = 1296, then 6n–3 = _________.
Proof: Let 6n=1296
⇒6n = 64
⇒n = 4
Now,
6n−3=64−3
=61
=6
Hence, if 6n = 1296, then 6n–3 = 6.
Q.6. The value of 4 × (256)–1/4 ÷ (243)1/5 is ________.
Proof: Let x=
⇒x =
⇒x =
⇒x =
⇒x =
⇒x = 1 ÷ 3
⇒x = 1/3
Hence, the value of 4 × (256)–1/4 ÷ (243)1/5 is
Q.7. If (6x)6=623, then x = ________.
Proof:
Hence, if
Q.8. = ___________.
Proof:
(using the identity: a2−b2=(a+b)(a−b))
Hence,
Q.9. then x =_________.
Proof:
⇒2p = x
⇒x = 2p
Hence, if then x=2p.
Q.10. If 5n+2 = 625, then (12n + 3)1/3 = _________.
Proof: Let 5n+2=625
⇒5n+2 = 54
⇒n+2 =4
⇒n = 2
Now,
= 3
Hence, if 5n+2 = 625, then (12n + 3)1/3 = 3.
Q.11. If= __________.
Proof:
=(a)−a
Hence,
Q.12. If=33 , then 5x + 6y = __________.
Proof: Given: =33
⇒3−5x = 33
⇒−5x = 3
⇒5x = −3 ...(1)
(729)y = 33
⇒(36)y = 33
⇒36y = 33
⇒6y = 3 ...(2)
Adding (1) and (2), we get
5x + 6y=−3 + 3
=0
Hence, 5x + 6y = 0.
Q.13. If 6x–y = 36 and 3x+y = 729, then x2 – y2 = _________.
Proof: Given: 6x−y = 36
and 3x+y = 729
6x−y = 36
⇒ 6x−y = 62
⇒x−y = 2 ...(1)
3x+y = 729
⇒3x+y = 36
⇒x+y = 6 ...(2)
Adding (1) and (2), we get
2x = 8
⇒x = 4
Substituting the value of x in (2), we get
4+y = 6
⇒y = 2
Now,
x2−y2 = 42−22
=16 − 4
=12
Hence, x2 – y2 = 12.
Q.14. equals __________.
Proof:
=(2)1/6
Hence, equals
Q.15. The productis equal to ________.
Proof:
= (2)1
= 2
Hence, the product is equal to 2.
Q.16. is equal to _________.
Proof:
= (3)−2
= 1/32
= 1/9
Hence, is equal to
Q.17. The value of (256)0.16 × (256)0.09 is ________.
Proof: (256)0.16 × (256)0.09 = (256)0.16+0.09
= (256)0.25
= (2)2
= 4
Hence, the value of (256)0.16 × (256)0.09 is 4.
Very Short Answer Type Questions(VSAQs)
Q.1. Write (625)−1/4 in decimal form.
Proof: We have to write (625)−1/4 in decimal form. So,
Hence the decimal form ofis
Q.2. State the product law of exponents.
Proof:
State the product law of exponents.
If is any real number and m ,n are positive integers, then
By definition, we have
am x an = am+n
Thus the exponent "product rule" tells us that, when multiplying two powers that have the same base, we can add the exponents.
Q.3. State the quotient law of exponents.
Proof: The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. If a is a non-zero real number and m, n are positive integers, then
We shall divide the proof into three parts
(i) when m > n
(ii) when m = n
(iii) when m < n
Case 1
when m > n
We have
Case 2
when m = n
We get
Cancelling common factors in numerator and denominator we get,
By definition we can write 1 as a0
Case 3
when m < n
In this case, we have
Hence, whether m > n,m = n or, m < n
Q.4. State the power law of exponents.
Proof: The "power rule" tell us that to raise a power to a power, just multiply the exponents.
If a is any real number and m, n are positive integers, then
We have,
factors
factors
Hence,
Q.5. If 24 × 42 =16x, then find the value of x.
Proof: We have to find the value of x provided 24 × 42 =16x
So,
By equating the exponents we get
Hence the value of x is .
Q.6. If 3x-1 = 9 and 4y+2 = 64, what is the value of x/y ?
Proof: We have to find the value offor
So,
By equating the exponent we get
Let’s take
By equating the exponent we get
By substituting x = 3, y = 1 we get 3 / 1
Hence the value of x/y is 3.
Q.7. Write the value of
Proof: We have to find the value of So,
By using law rational exponentswe get
Hence the value ofis
Q.8. Writeas a rational number.
Proof: We have to find the value of
So,
Hence the value of the value ofis.
Q.9. Write the value of
Proof: We have to find the value of So,
= = 5×3 =15
Hence the value of the value of is
Q.10. For any positive real number x, find the value of
Proof: We have to find the value of L =
By using rational exponentswe get
By using rational exponentswe get
By definition we can write x0 as 1
Hence the value of expression is.
Q.11. Write the value of
Proof: We have to find the value of So,
By using rational exponents we get
Hence the simplified value ofis.
Q.12. Simplify
Proof: We have to simplify So,
Hence, the value ofis .
Q.13. For any positive real number x, write the value of
Proof: We have to simplifySo,
By using rational exponents, we get
Hence the value ofis x3.
Q.14. If (x − 1)3 = 8, What is the value of (x + 1)2 ?
Proof: We have to find the value of (x + 1)2, where (x − 1)3 = 8
Consider (x − 1)3 =23
By equating the base, we get
x - 2 = 1
x = 2 + 1
x = 3
By substituting x = 3 in (x + 1)2
= (x + 1)2
= (3 + 1)2is
= 42
= 16
Hence the value of(x + 1)2 is 16.
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1. What are exponents of real numbers? |
2. How do you simplify expressions with exponents? |
3. What is the meaning of a negative exponent? |
4. Can exponents be applied to any real number? |
5. How are exponents used in real-life situations? |
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