\(5^3\) is a power with base \(5\) and exponent \(3.\)
\(5^{3}\) means \(5\times 5\times 5\) and is equal to \(125\) when expressed as an integer.
When \(8^{674}\times 5^{2025}\) is expressed as an integer, how many digits are in the product?
An immediate temptation might be to reach for a calculator. In this case, basic calculator technology will let you down. We will solve this problem using our knowledge of powers and corresponding power laws. \[\begin{aligned} 8^{674}\times 5^{2025}&= \left((2^3)^{674}\right)\times 5^{2025}\\ &=2^{3\times 674}\times 5^{2025}\\ &=2^{2022}\times 5^{2025}\\ &=2^{2022}\times 5^{2022}\times 5^3\\ &=(2\times 5)^{2022}\times 125\\ &=10^{2022}\times 125 \end{aligned}\] But \(10^{2022}\) is the number \(1\) followed by \(2022\) zeroes. When we multiply this number by the three-digit number \(125\), we obtain the number \(125\) followed by \(2022\) zeroes. Therefore, \(8^{674}\times 5^{2025}\) has \(2022+3=2025\) digits. Happy New Year again!