Question

What is the value of \( \left( \frac{3^{81}}{27^4} \right)^{\frac{1}{3}} \)?

• A. \( 3^{13} \)
• B. \( 3^{96} \)
• C. \( 3^{23} \)
• D. \( 3^{69} \)

Hint:

When simplifying exponential expressions, express all terms with the same base, apply exponent laws (like \( a^m / a^n = a^{m-n} \)), and simplify powers systematically.

Answer 1

The Correct Option is C

We are given: \[ \left( \frac{3^{81}}{27^4} \right)^{\frac{1}{3}} \] First, express 27 in terms of base 3: \[ 27 = 3^3 \Rightarrow 27^4 = (3^3)^4 = 3^{12} \] Now substitute: \[ \left( \frac{3^{81}}{3^{12}} \right)^{\frac{1}{3}} = \left( 3^{81 - 12} \right)^{\frac{1}{3}} = \left( 3^{69} \right)^{\frac{1}{3}} = 3^{69 \div 3} = 3^{23} \]