Given that two to the 𝑥th power is
equal to 32, find the value of 𝑥.
Here we have an equation involving
an exponent which is a variable. The exponent here is equal to
𝑥. Now, usually, to solve an equation,
we’d look to perform a series of inverse operations, but the inverse to finding the
𝑥th power is to find the 𝑥th root, which doesn’t really help us much. Instead, it’s worth noticing that
32 can be written as a power of two. In fact, we know that two to the
fifth power is equal to 32. And this means we can rewrite our
equation as two to the 𝑥th power equals two to the fifth power.
So, how does this help? Well, now the base — that’s the big
number; here that’s two — is the same. And so we can say that for this
equation to make sense, the exponents must also be the same. That is, 𝑥 is equal to five. Now, whenever we’re solving
equations, it always makes sense to check our answer by substituting it back into
the original equation. Let’s let 𝑥 be equal to five. And then two to the 𝑥th power
becomes two to the fifth power, which is indeed equal to 32. And so, given that two to the 𝑥th
power is equal to 32, 𝑥 must be equal to five.