Question

If \( 10^k = \frac{128 \times (16 \times 10^{-4})^3{2^{19} \times 10^{-11}} \), then the value of \( k \) is:}

• A. 2
• B. 1
• C. 0
• D. -1

Hint:

When solving exponential equations with different bases, express each part in terms of powers of 10 and simplify step by step.

Answer 1

The Correct Option is D

Step 1: Calculate the cube of \( 16 \times 10^{-4} \). Since \( 16 = 2^4 \), we have: \[ (16 \times 10^{-4})^3 = (2^4 \times 10^{-4})^3 = 2^{12} \times 10^{-12} = 4096 \times 10^{-12} \] Step 2: Simplify the denominator. \( 2^{19} \times 10^{-11} = 524288 \times 10^{-11} \) Step 3: Set up the fraction and simplify. \[ \frac{128 \times 4096 \times 10^{-12}}{524288 \times 10^{-11}} = \frac{524288 \times 10^{-12}}{524288 \times 10^{-11}} = \frac{10^{-12}}{10^{-11}} = 10^{-1} \] Step 4: Solve for \( k \). \[ 10^k = 10^{-1} \] Hence, \( k = -1 \).