Write four to the second power times nine to the fifth power times four to the third power times nine to the fourth power in the form 𝑎 to the 𝑚 power 𝑏 to the 𝑛 power, where 𝑎 and 𝑏 are prime numbers.
So we want to take this expression and put it into this form: 𝑎 to the 𝑚 power 𝑏 to the 𝑛 power. But we want 𝑎 and 𝑏 to be prime. Prime numbers are only divisible by itself and one, meaning they only have two factors — two numbers that multiply to be that exact number.
Some prime numbers would be two, three, five, seven, 11 because they’re only divisible by themselves and one. So two is only divisible by itself and one, so is three. Four is not prime because four is divisible by two. We have five that’s prime. Six is not prime because it’s divisible by three and two.
We have seven, eight, nine, and 10 are not prime because they are divisible by other numbers and so on. So the bigger numbers at the bottom are the bases and then the small numbers at the top are exponents. So we have two bases that are four and two bases that are nine. Neither one of those numbers is prime.
However, we could rewrite four so that way it has a base that is prime because four is the same thing as two squared. And in that case, the base would be prime. So let’s change the fours to be two squared. Okay, so we’ve done so. Now, let’s also change the nines. Nine we could rewrite as three squared because three is prime. That’s why we want to use three.
So now, we have to remember a rule about exponents. When we have 𝑎 to the 𝑚 power that’s raised to the 𝑛 power, that’s equal to 𝑎 to the 𝑚 times 𝑛. So essentially, we just multiply those exponents together. And that’s what we have here. Therefore, we would have two to the fourth power times three to the 10th power times two to the sixth power times three to the eighth power.
Now, another exponent rule we need to remember is that if we have 𝑎 to the 𝑚 power times 𝑎 to the 𝑛 power, that’s equal to 𝑎 to the 𝑚 plus 𝑛 power. So we add the exponents. So if we write the bases of twos next to each other and the bases of threes next to each other, it’s a little little easier to see this.
So we need to add four and six because they both have a base of two. So we have two to the 10th power. And then, we need to add 10 and eight. So we have three to the 18th power. So we have rewritten our expression, where the bases are prime numbers: two to the 10th power three to the 18th power.
We also could have gotten our answer another way using the same rules about exponents. So originally, we could have rewritten it so the fours are next to each other. And then, maybe we would have noticed right from here that with fours being the same base, we can add their exponents. So we would have four to the fifth power and nine to the ninth power.
Then, we could have replaced our bases with something different. So instead of four, we wanna a prime number and instead of nine, we wanna a prime number. So instead of four, we replace it with two squared and instead of nine, we replace it with three squared.
And here is where we multiply our exponents. Two times five gives us 10 and two times nine gives us 18, therefore, resulting in the same answer: two to the 10th power three to the 18th power.