Write 343 squared times three cubed times 343 to the fourth power times three to the fourth power in the form 𝑎 to the 𝑚 power 𝑏 to the 𝑛 power where 𝑎 and 𝑏 are prime numbers.
Taking this expression, we might first want to simplify and regroup so that we have three cubed times three to the fourth power times 343 squared times 343 to the fourth power. We know that 𝑎 to the 𝑥 power times 𝑎 to the 𝑦 power is equal to 𝑎 to the 𝑥 plus 𝑦 power. And so, we could take one additional step and simplify these values. Three cubed times three to the fourth power equals three to the three plus four power, three to the seventh power. And we take 343 to the two plus four power, 343 to the sixth power.
It looks like we’re close to the form 𝑎 to the 𝑚 power times 𝑏 to the 𝑛 power. But remember, 𝑎 and 𝑏 should be prime. Now, three is prime. So 𝑎 to the 𝑚 power can be three to the seventh power, but 343 is not prime. And that means we need to do some more simplification. For now, we’ll say that we have one correct term, and we want the prime factors for 343. Now, 343 is not divisible by two because it’s not even. Therefore, it’s not divisible by four, six, eight, or 10. To find out if it’s divisible by three, we can add three plus four plus three, which equals 10. 10 is not a multiple of three. Therefore, 343 is not divisible by three. 343 does not end in a five or zero, so it’s not divisible by five.
There’s no simple way to check for divisibility by seven. So we need to divide. We want to know, does seven go into 343 evenly? Seven goes into 34 four times. Four times seven is 28. 34 minus 28 is six. And seven goes into 63 nine times, with a remainder of zero. This means 343 is equal to seven times 49. And we recognize 49 as seven squared. So we could say 343 equals seven times seven times seven or seven cubed. We know that seven is a prime number. And so, we can substitute seven cubed in place of 343 in our expression. And we’ll have seven cubed to the sixth power instead of 343 to the sixth power.
We remember 𝑎 to the 𝑥 power to the 𝑦 power is equal to 𝑎 to the 𝑥 times 𝑦 power. That’s a power of a power. And that means seven cubed to the sixth power will be equal to seven to the three times six power, seven to the 18th power. We bring down the first term we found, three to the seventh power. And we have two prime bases. And so, we could say three to the seventh power times seven to the 18th power or seven to the 18th power times three to the seventh power. Both options fit the format we were given.