Solve for x: 1/x³ − 1/(5 × 10⁻⁵)³ = 9.33020554 × 10¹⁰.

Step-by-step solution

1. 1

   Evaluate the known reciprocal cube. Cube the mantissa and multiply the exponent by three:

   $\left(5\times 10^{-5}\right)^3 = 125\times 10^{-15} = 1.25\times 10^{-13} \ \Longrightarrow \ \frac{1}{1.25\times 10^{-13}} = 8\times 10^{12}$

2. 2

   Move it across and match the powers of ten before adding.

   $\frac{1}{x^3} = 8\times 10^{12} + 9.33020554\times 10^{10}$

   Rewrite the larger term on the smaller exponent:

   $8\times 10^{12} = 800\times 10^{10}$

   $\frac{1}{x^3} = (800 + 9.33020554)\times 10^{10} = 809.33020554\times 10^{10} = 8.0933020554\times 10^{12}$

3. 3

   Invert to get x³.

   $x^3 = \frac{1}{8.0933020554\times 10^{12}} = 1.235591\times 10^{-13}$

4. 4

   Take the cube root, shifting to an exponent divisible by 3.

   $x = \sqrt[3]{123.5591\times 10^{-15}} = \sqrt[3]{123.5591}\times 10^{-5} = 4.98071\times 10^{-5}$

5. 5

   Do not round this to 5×10⁻⁵ — check why. The answer is temptingly close to the constant in the problem, but $x = 5\times 10^{-5}$ exactly would give

   $\frac{1}{(5\times10^{-5})^3} - \frac{1}{(5\times10^{-5})^3} = 0$

   not $9.33\times 10^{10}$. The two are different by construction, and rounding away the difference destroys the equation entirely.

6. 6

   Verify the answer and see why the shift is so small. With $x = 4.98071\times 10^{-5}$:

   $\frac{1}{x^3} - 8\times 10^{12} = 9.3302\times 10^{10} \ \checkmark$

   The right-hand side is only $1.17\%$ of $8\times 10^{12}$, and since $x \propto (1/x^3)^{-1/3}$, a $1.17\%$ change in $1/x^3$ moves $x$ by roughly $1.17/3 \approx 0.39\%$ — from $5.0000\times10^{-5}$ down to $4.9807\times10^{-5}$.