Find the Value of the Exponent Expression

Let's find the value of the given mathematical expression involving exponents: \[ \left[(3 \times 3 \times 3 \times 3 \times 3 \times 3)^6 \div (3 \times 3 \times 3 \times 3)^7 \times 3^4\right] \] To solve this, we will use the fundamental rules of exponents.

Understanding Exponent Rules

Before we dive into the calculation, let's recall the key exponent rules that will be applied:

• Product Rule: When multiplying powers with the same base, add the exponents. If \(a\) is a real number and \(m, n\) are integers, then \(a^m \times a^n = a^{m+n}\).
• Quotient Rule: When dividing powers with the same base, subtract the exponents. If \(a\) is a real number and \(m, n\) are integers, then \(a^m \div a^n = a^{m-n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents. If \(a\) is a real number and \(m, n\) are integers, then \((a^m)^n = a^{m \times n}\).

Step-by-Step Calculation of the Expression Value

We will simplify the given exponent expression step by step:

1. Simplify terms inside parentheses: First, express the repeated multiplications as powers of 3.
   • The term \(3 \times 3 \times 3 \times 3 \times 3 \times 3\) is equal to \(3^6\).
   • The term \(3 \times 3 \times 3 \times 3\) is equal to \(3^4\).
   Substitute these back into the original expression: \[ \left[(3^6)^6 \div (3^4)^7 \times 3^4\right] \]
2. Apply the Power of a Power Rule: Next, simplify the terms where a power is raised to another power using the rule \((a^m)^n = a^{m \times n}\).
   • For \((3^6)^6\), we multiply the exponents: \(3^{6 \times 6} = 3^{36}\).
   • For \((3^4)^7\), we multiply the exponents: \(3^{4 \times 7} = 3^{28}\).
   The expression now becomes: \[ \left[3^{36} \div 3^{28} \times 3^4\right] \]
3. Perform the Division: Apply the Quotient Rule \(a^m \div a^n = a^{m-n}\) to the division part of the expression. \[ 3^{36} \div 3^{28} = 3^{36 - 28} = 3^8 \] The expression is now: \[ 3^8 \times 3^4 \]
4. Perform the Multiplication: Apply the Product Rule \(a^m \times a^n = a^{m+n}\) to the remaining multiplication. \[ 3^8 \times 3^4 = 3^{8 + 4} = 3^{12} \] So, the simplified value of the expression is \(3^{12}\).
5. Express in terms of 27: The options might present the answer in a different base. We know that \(27 = 3 \times 3 \times 3 = 3^3\). We can rewrite \(3^{12}\) using \(27\) as the base. \[ 3^{12} = 3^{3 \times 4} = (3^3)^4 \] Substitute \(3^3 = 27\): \[ (3^3)^4 = 27^4 \]

Therefore, the value of the given expression is \(27^4\).