If 45𝑥 to the power 𝑚 𝑦 to the
power 𝑚 plus 𝑛 and 35𝑥 to the power 29 𝑦 to the power 32 are like terms, what
are the values of 𝑚 and 𝑛?
In this question, we’re considering
like terms. If we have two terms that are
described as like terms, this means that the exponents of the variables must be
equal to each other. The coefficients of the numbers
that we multiply by can be different. So in this case, we can effectively
ignore our coefficients 45 and 35 and focus simply on our variables 𝑥 and 𝑦. So let’s begin by considering our
If we’re told that they’re like
terms, this means that 𝑥 to the power of 𝑚 must be equal to 𝑥 to the power of
29. So since the exponents must be
equal, we can then say that 𝑚 equals 29. I’m moving on for our 𝑦
variable. If they’re like terms, then 𝑦 to
the power of 𝑚 plus 𝑛 must be equal to 𝑦 to the power of 32. And we can put those exponents
equal, which means that 𝑚 plus 𝑛 must be equal to 32.
And as we’ve already found the
value 𝑚 equals 29, we can substitute that into our equation. And in order to find 𝑛 by itself,
we subtract 29 from both sides of the equation, giving us 𝑛 equals 32 minus 29. So 𝑛 equals three.
And so, our final answer is 𝑚
equals 29 and 𝑛 equals three.