show that x3 − 6x = 6
x3 − 6x = 6
Putting cube on both the sides, we get
As we know, (a+b)3 = a3 + b3 + 3ab(a+b)
x3 = 6 + 6x
x3 – 6x = 6
Hence proved
12. Determine (8x)x, if 9x+2 = 240 + 9x.
9x+2 = 240 + 9x
9x .92 = 240 + 9x
Let 9x be y
81y = 240 + y
81y – y = 240
80y = 240
y = 3
Since, y = 3
Then,
9x = 3
32x = 3
Therefore, x = 1/2
= 2
Therefore (8x)x = 2
13. If 3x+1 = 9x-2, find the value of 21+x
3x+1 = 9x-2
3x+1 = 32x-4
x + 1 = 2x – 4
x = 5
Therefore the value of 21+x = 21+5 = 26 = 64
14. If 34x = (81)-1 and (10)1/y = 0.0001, find the value of 2-x+4y.
34x = (81)-1 and (10)1/y = 0.0001
34x = (3)-4
x = -1
And (10)1/y = 0.0001
(10)1/y = (10)−4
To find the value of 2-x+4y, we need to substitute the value of x and y
15. If 53x = 125 and 10y = 0.001. Find x and y.
53x = 125 and 10y = 0.001
53x = 53
x = 1
Now,
10y = 0.001
10y = 10-3
y = -3
Therefore, the value of x = 1 and the value of y = -3
16. Solve the following equations
(i) 3x+1 = 27×34
3x+1 = 33×34
3x+1 = 33+4
x + 1 = 3 + 4 [By equating exponents]
x + 1 = 7
x = 7 − 1
x = 6
24x = 23
4x = 3 (By equating exponents)
x = 3/4
(By equating exponents)
(iii). 3x−1×52y−3 = 225
3x−1×52y−3 = 32×52
x−1 = 2 [By equating exponents]
x = 3
3x−1×52y−3 = 32×52
2y−3 = 2 [By equating exponents]
2y = 5
y = 5/2
3x+3 = 4y+8 —eq1
3(3x+3) = 6+2x+24
9x+9 = 30+2x
7x = 21
x = 21/7
x = 3
Putting value of x in eq2
2x−2−3+2x = −3x [By equating exponents]
4x+3x = 5
7x = 5
x = 5/7
[By equating exponents]
17. If a and b are distinct positive primes such that find x and y
18.If a and b are different positive primes such that
axby find x and y
(ii) (a+b)−1(a−1+b−1) = axby,find x and y
=(ab)−1 = a−1b−1
By equating exponents
x = −1,y = −1
Therefore x+y+2 = −1−1+2 = 0
19. If 2x×3y×5z = 2160 , find x,y and z. Hence compute the value of 3x×2−y×5−z
20. 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
We have to find the value of
= 24/49
21. Simplify
22. Show that
Hence LHS = RHS
23.
(i)
= x0 y0
= 1
1. What are exponents and how are they used in real numbers? |
2. How do exponents affect the value of a real number? |
3. What is the significance of the exponent 0 in real numbers? |
4. How can exponents be used to simplify calculations with real numbers? |
5. Can exponents be used with all types of real numbers? |
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