Questions: Simplify. Express your answer using exponents. (12a^3b^7c^5)^2

Solution Steps

To simplify the expression \((12 a^{3} b^{7} c^{5})^{2}\), we need to apply the power of a power property of exponents. This property states that \((x^m)^n = x^{m \cdot n}\). We will apply this property to each component inside the parentheses: the constant 12 and the variables \(a^3\), \(b^7\), and \(c^5\).

To simplify the expression \((12 a^{3} b^{7} c^{5})^{2}\), we apply the power of a power property of exponents, which states that \((x^m)^n = x^{m \cdot n}\). This means we will square each component inside the parentheses: the constant \(12\) and the variables \(a^{3}\), \(b^{7}\), and \(c^{5}\).

Calculating each component, we have:

• For the constant: \[ 12^{2} = 144 \]
• For the variable \(a\): \[ (a^{3})^{2} = a^{3 \cdot 2} = a^{6} \]
• For the variable \(b\): \[ (b^{7})^{2} = b^{7 \cdot 2} = b^{14} \]
• For the variable \(c\): \[ (c^{5})^{2} = c^{5 \cdot 2} = c^{10} \]

Combining all the results, we obtain the simplified expression: \[ 144 a^{6} b^{14} c^{10} \]

Final Answer