# Using differentials, find the approximate value of each of the following

(i) (17/81)^{1/4 }(ii) (33)^{1/5}

**Solution:**

We can use differentials to calculate small changes in the dependent variable of a function corresponding to small changes in the independent variable

(i) (17/81)^{1/4}

Let us consider y = (x)^{1/4}

Assume x = 16/81 and Δx = 1/81

Then,

Δy = (x + Δx)^{1/4} - (x)^{1/4}

= (17/81)^{1/4} + (16/81)^{1/4}

= (17/81)^{1/4} - 2/3

2/3 + Δy = (17/81)^{1/4}

Now, dy is approximately equal to Δy and is given by,

dy = (dy/dx)Δx

= 1/4(x)^{3/4} Δx

[∵ y = (x)^{1/4}]

= 1/4(16/81)^{3/4} (1/81)

= 27/(4 x 8) x (1/81)

= 1/(32 x 3)

= 1/96

= 0.010

Hence,

(17/81)^{1/4} = 2/3 + 0.010

= 0.667 + 0.010

= 0.677

Thus, the approximate value of (17/81)^{1/4} is 0.677.

(ii) (33)^{1/5}

Consider, y = (x)^{1/5}

Let, x = 32 and Δx = 1

Then,

Δy = (x + Δx)^{1/5} - (x)^{1/5}

= (33)^{1/5} + (32)^{1/5}

= (33)^{1/5} - 1/2

1/2 + Δy = (33)^{1/5}

Now, dy is approximately equal to Δy and is given by,

dy = (dy/dx)Δx

= - 1/5(x)^{6/5} Δx

[since y = (x)^{1/5}]

= - 1/5(2)^{6/5} (1)

= - 1/320

= - 0.003

Hence,

(33)^{1/5} = 1/2 + (- 0.003)

= 0.5 - 0.003

= 0.497

Thus, the approximate value of (33)^{1/5} is 0.497

NCERT Solutions Class 12 Maths - Chapter 6 Exercise ME Question 1

## Using differentials, find the approximate value of each of the following (i) (17/81)^{1/4 }(ii) (33)^{1/5}

**Summary:**

Using differentials, the approximate value of (i) (17/81)^{1/4 }is 0.677 and the approximate value of (ii) (33)^{1/5} is 0.497

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