Video Transcript
Evaluate 0.0625 to the power
0.25.
Our approach to evaluating this
will involve taking both of our decimals, the base and the exponent, and writing
those as fractions. So 0.0625 is equivalent to 625 over
10000, and our exponent of 0.25 is equivalent to one-quarter. We can then apply some exponent
rules.
The first rule we’re going to use
is that if we have a fraction 𝑥 over 𝑦 to the power of 𝑎, this is equivalent to
𝑥 to the power of 𝑎 over 𝑦 to the power of 𝑎. So our fraction is equivalent to
625 to the power of one-quarter over 10000 to the power of one-quarter.
Now let’s think about what it means
to be to the power of one-quarter. We can use our second rule to help
us here, which says that if we have a value 𝑥 to the power of one over 𝑎, it’s
equivalent to the 𝑎th root of 𝑥. So our fractional exponent of
one-quarter is equivalent to the fourth root.
On the numerator then, we have the
fourth root of 625. And on the denominator, it’s the
fourth root of 10000. Evaluating the fourth root of 625
gives us five, since if we write five down four times and multiply, we’ll get
625. And then the fourth root of 10000
is 10 since again if we write down 10 four times and multiply, we get 10000. We can then simplify our fraction
five-tenths, giving us a final answer of a half.
