Question 7 Simplify completely: \( \left(5^{12} \cdot 5^{-3}\right)^{6}\left(25^{4}\right) \) Base \( = \) Blank 1 Exponent \( = \) Blank 2

To simplify the expression \( \left(5^{12} \cdot 5^{-3}\right)^{6}\left(25^{4}\right) \), we start by handling the first part. Using the properties of exponents, we can combine the bases: \[ 5^{12} \cdot 5^{-3} = 5^{12-3} = 5^{9} \] Now, we raise this to the 6th power: \[ \left(5^{9}\right)^{6} = 5^{9 \cdot 6} = 5^{54} \] Next, we address \( 25^{4} \). Since \( 25 \) is \( 5^{2} \): \[ 25^{4} = (5^{2})^{4} = 5^{2 \cdot 4} = 5^{8} \] Now we can combine \( 5^{54} \) and \( 5^{8} \): \[ 5^{54} \cdot 5^{8} = 5^{54 + 8} = 5^{62} \] Thus, the final answer for the base and exponent is: Base \( = 5 \) Exponent \( = 62 \) --- In the realm of mathematics, exponents are like the superheroes of arithmetic—they amplify values in a flash! Knowing how to manipulate them can make problem-solving much easier and save you time. Just keep in mind that any positive number raised to zero equals one, which is a fun little rule to remember for those moments when you encounter zero exponents. Another handy tip is to always look out for common bases when multiplying or dividing—this helps you streamline your calculations. For example, combining bases like \(5^{12}\) and \(5^{-3}\) not only lightens the workload but can also help you see the relationships between numbers more clearly—plus, it’s a great way to impress friends with your exponent skills!