Express 𝑥 to the power of 16 over
three multiplied by 𝑥 to the power of 23 over two in the form the
[𝑚th] root of 𝑎 to the power of 𝑛.
To solve this problem, we have to
write some rules of exponents that’ll help us out. So if we’re gonna express 𝑥 to the
power of 16 over three multiplied by 𝑥 to the power of 23 over two, the first rule
we’re gonna use is this one. Our first rule we are looking at is
that 𝑎 to the power of 𝑚 multiplied by 𝑎 to the power of 𝑛 is equal to 𝑎 to the
power of 𝑚 plus 𝑛.
So let’s apply this to our
question; that’s gonna give us 𝑥 to the power of 16 over three plus 23 over
two. And to add these powers, we’re
gonna make the denominators equal. So we’re gonna have 𝑥 to the power
of- we get 23 over six. And we get that by multiplying the
numerator and the denominator both by two. And we do that because six is the
lowest common multiple of two and three, which are both for our denominators.
And then we’re gonna have plus 69
over six, again this time multiplying the numerators on it by three to give us that
denominator of six. And then when we add the powers
together, we get our final answer, which is 𝑥 to the power of 101 over six. But is this it? Is this the problem solved? That’s actually a no because having
a look at the original question, we can see that we need to leave it in this
particular form. As to be able to do this, what
we’re gonna do is actually use another rule of exponents. And this rule of exponents shows us
that 𝑎 to the power of 𝑛 over 𝑚 is equal to the 𝑚th root of 𝑎 to the power of
Okay, great! Let’s apply this to the term we
have. Well first of all, we know that is
going to be sixth root because if we look at our rule of exponents we have that the
bottom of the denominator, so the 𝑚 in this case, is equal to six in our term. And then our power of 𝑥 is gonna
be 101 because again looking at this, this is the numerator, so our 𝑛 value if you
look at the rule of exponents. It’s there we have it: 𝑥 to the
power of 16 over three multiplied by 𝑥 to the power of 23 over two is equal to the
sixth root of 𝑥 to the power of 101.