The approximate value of `(7.995)^(1//3)` correct to four decimal
places , is

Options

A. 1.9995
B. 1.9996
C. 1.999.0
D. 1.9991

Answer

To find the approximate value of \( (7.995)^{1/3} \) correct to four decimal places, we can use the Binomial Theorem. Here’s the step-by-step solution:

### Step 1: Rewrite the expression
We start by rewriting \( 7.995 \) in a form that is easier to work with:
\[ 7.995 = 8 - 0.005 \]
Thus, we can express \( (7.995)^{1/3} \) as:
\[ (7.995)^{1/3} = (8 - 0.005)^{1/3} \]

**Hint:** Look for a nearby perfect cube to simplify the calculation.

### Step 2: Factor out the cube root of 8
We can factor out \( 8 \) from the expression:
\[ (8 - 0.005)^{1/3} = 8^{1/3} \left(1 - \frac{0.005}{8}\right)^{1/3} \]
Since \( 8^{1/3} = 2 \), we have:
\[ = 2 \left(1 - \frac{0.005}{8}\right)^{1/3} \]

**Hint:** Recognize that \( 8^{1/3} \) simplifies to a whole number.

### Step 3: Simplify the fraction
Now we simplify \( \frac{0.005}{8} \):
\[ \frac{0.005}{8} = 0.000625 \]
So, we rewrite our expression:
\[ = 2 \left(1 - 0.000625\right)^{1/3} \]

**Hint:** Simplifying fractions can help in reducing complexity.

### Step 4: Apply the Binomial Theorem
Using the Binomial expansion for \( (1 + b)^n \), where \( n = \frac{1}{3} \) and \( b = -0.000625 \):
\[ (1 - 0.000625)^{1/3} \approx 1 + \frac{1}{3}(-0.000625) \]
Calculating this gives:
\[ \approx 1 - \frac{0.000625}{3} = 1 - 0.0002083333 \approx 1 - 0.000208 = 0.9997916667 \]

**Hint:** The first term of the binomial expansion often gives a good approximation.

### Step 5: Combine the results
Now, substituting back into our expression:
\[ (7.995)^{1/3} \approx 2 \times 0.9997916667 = 1.9995833334 \]

**Hint:** Multiplying the approximated value by the factor obtained earlier is crucial.

### Step 6: Round to four decimal places
Finally, we round \( 1.9995833334 \) to four decimal places:
\[ \approx 1.9996 \]

Thus, the approximate value of \( (7.995)^{1/3} \) correct to four decimal places is:
\[ \boxed{1.9996} \]