Calculate the square root of nine to the fourth power times nine to the second power divided by nine to the third power times nine to the fifth power.
There are a few properties that we can use here because we will be working with like basis of nine. So when multiplying with like basis, we can add our exponents together. And when dividing with like basis, we can subtract their exponents together. So on the numerator, we can add the four and two together. And on the denominator, we can add the three and five together. So we have the square root of nine to the sixth divided by nine to the eighth.
So with the nines, we are dividing like basis of nine. So we can subtract their exponents of six and eight. So we have the square root of nine to the six minus eight power. So, we have the square root of nine to the negative second power. From here, we can replace the square root symbol with the power of one-half. And we can use the property where when we have 𝑥 to the 𝑎 to the 𝑏 power, so a power raised to another power, we actually multiply our powers together. So we need to multiply negative two and one-half together.
So our power is negative two divided by two, which is equal to negative one. And nine to the negative first power is equal to one-ninth because when we have numbers raised to a negative exponent, if it’s on a numerator, we can move it to the dominator and make the power positive. Or, if it’s on the dominator and the power is negative, we can move it to the numerator and make the power positive.
We also could’ve worked a little bit differently from here. Nine to the fourth times nine squared means that there are four nines times two nines. So, all together, there are six nines. And then, for the denominator, nine to the third times nine to the fifth means there’s three nines and five nines, which makes sense because we have nine to the sixth power and nine to the eighth power. So all of the nines on the numerator cancel. So on the numerator we have one. Now, on the denominator, there’s two nines, so nine times nine, which is 81. So we can separate the numerator and denominator as separate square roots. So the numerator, square root of one is one. And the square root of 81 is nine. So, once again, we get one-ninth. So one-ninth will be our final answer.