Power of five Powers with brackets, addition and multiplication Age 14 to 16 Challenge level Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving Being curious Being resourceful Being resilient Being collaborative Problem Student Solutions Problem Find the value of $x$ so that:5(5x+5x+5x+5x+5x)2=59This problem is adapted from the World Mathematics Championships Student Solutions Answer: $x=3$Rearranging the equation5(5x×5)2=59⇒(5x×5)2=58⇒5x×5=58=54⇒5x=53⇒x=3Using index laws firstFirst, look at what is contained inside the brackets:5x+5x+5x+5x+5x=5×5x=51×5x=51+x So: 5(5x+5x+5x+5x+5x)2=5(51+x)2=5×5(1+x)×2=51×52+2x=51+2+2x=53+2x So $5^{3+2x}=5^9$, which means $3+2x=9$, so $2x=6$, so $x=3$.