Solving for the Unknown Exponent

The question requires finding the value of the exponent represented by the question mark (?) in the given equation:

To solve this, we need to simplify the left side of the equation using the laws of exponents, expressing each term as a power of 12.

Simplifying the Equation Components

• First term: $ (12\times12)^3 = (12^2)^3 $. Using the power of a power rule $ (a^m)^n = a^{m \times n} $, this becomes $ 12^{2 \times 3} = 12^6 $.
• Second term: $ (144\times12)^3 $. Since $ 144 = 12^2 $, the term is $ (12^2 \times 12^1)^3 $. Using the product of powers rule $ a^m \times a^n = a^{m+n} $, this simplifies to $ (12^{2+1})^3 = (12^3)^3 $. Applying the power of a power rule again gives $ 12^{3 \times 3} = 12^9 $.
• Third term: $ (20736)^2 $. We need to express 20736 as a power of 12. $ 12^4 = 20736 $. So, the term is $ (12^4)^2 $. Using the power of a power rule, this becomes $ 12^{4 \times 2} = 12^8 $.

Applying Exponent Laws

Substitute the simplified terms back into the original equation:

Now, apply the division and multiplication rules for exponents:

• Division: $ a^m \div a^n = a^{m-n} $. So, $ 12^6 \div 12^9 = 12^{6-9} = 12^{-3} $.
• Multiplication: $ a^m \times a^n = a^{m+n} $. Applying this to the result: $ 12^{-3} \times 12^8 = 12^{-3+8} = 12^5 $.

Determining the Final Exponent

The simplified equation is:

By equating the exponents on both sides, we find:

Thus, the value that should come in place of the question mark is 5.