Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Write an equivalent expression using radical notation and if possible simplify. So for this we know that the exponent is 1 and the radical is 7. So the equivalent is just going to be the 7th root of Y to the first, or just the 7th root of Y. This next one, it's going to be the cube root of 8. But we know that 8 is just really 2 * 2 * 2. So the cube root of 2 ^3 would give us 2, this next one the 64 to the one 6th. So we're going to write 6 here. But 64 if we thought about prime factoring it, that's 8 * 8 2 * 4 two times 4. Again, those 4 split up, so we really get 2 to the 6th. So the 6th root of 2 to the 6th simplifies to two AB to the 1/4, just the 4th root of ABB to the three halves sqrt b ^3. The number in the bottom is the index. Remember here sqrt 4 to the 7th. Now this one we can simplify because we can actually find sqrt 4, sqrt 4 we know is 2 and 2 to the 7th. 248-1630 260-4128 24 8163264 one 28 So we can do this in any order we want. Actually we could have done 4 to the 7th and then taken the square root of it. But 4 to the 7th is really, really big. Sqrt 4 we knew easily. 2 to the 7th we just double and double and double. So 81 to the three halves. Sqrt 81 ^3 Well, we know sqrt 81 it's 9 and 9 ^3 9 * 9 is 8181 * 9 is 7/29. 1:25 A to the 2/3. So the cube root of 125 A and then that's squared. Well, we know the cube root of 125 is five, and then we need to square that. We don't know the cube root of A, so we're going to leave it underneath. 5 ^2 is 25, so we get 25, the cube root of a ^2. This next one, what we're really doing on this last one and this one is we're going to think about having the 9 by itself and then the Y to the 6th by itself. So if we have sqrt 9 ^3 and the square root of Y to the 6th cubed, sqrt 9 is 3/3 cubed is just going to give us 27 sqrt y to the 6th. Well, two goes in the sixth three times, so we're going to get Y to the third, but then we need to cube it, so we're going to get 27 Y to the 9th. Is that final answer there? This next type is we want to write an equivalent expression using exponential. So 19 to the one third power, A to the five halves power. Remember, if there's not a number, it's understood to be 211 to the one half power, N to the seven 6th power, X to the one fifth, Y to the one fifth. Sometimes you might see an intermediate step saying XY to the 1/5. But if we have an X and AY multiplied to the one fifth, then each of them go to that power separately. We get X to the 4 sevenths, Y to the 5 sevenths, and Z to the three sevenths. Just the numerator over the index here we're going to get 7 to the Four Thirds, X to the Four Thirds, Y to the Four Thirds. The four exponent is on all three of them, and the index of three is on all three of them. So all three of those get to the 4 thirds, three a / C to the 2/5. This 9 is on everything. The 11 is on everything, so 2 to the 9 elevenths X powers to powers we multiply so 6 * 954 / 11 and Y to the 9 elevenths. This next one we want to write an equivalent with only positive exponents. So if we have a negative exponent, it just means one over it. And if the negative exponent happens to be in the bottom, it just means we move it to the top because we could think of it as one over it. But one over something over one, one over one over something is really just X to the north. So this first example is just going to be 1 / y to the one fifth. This next one one over three X ^2 y to the positive 3/4. This next one, it's already a fraction, so we can actually think about just taking the reciprocal of it. So instead of 1 / 7, it's going to be 7 / 1 to the positive 3/4. This a was in the denominator, so we're just going to move it up to the top. So A to the 3/5 5 has a positive exponent. The X to the -2 thirds is going to go to the bottom. The 4/5 was positive and the Z was positive. So those are all staying up on top. This next one to Z, the five is a positive exponent, understood of one, but that X to the negative 1/3 is going to move up on top to make it X to the positive 1/3, 1 / 8 to the negative 2/3. That's going to turn into eight to the 2/3. But we can actually evaluate this one. What's the cube root of 8? Oops. Cube root of 8. Cube root of 8 is just two, right? And then we're going to square that. So 2 ^2 is going to give us 4. This next one to A to the 2/3, the B is a negative exponent, so it moves down below. The C is positive so it stays where it was. Finally, a negative of a fraction. Literally we just flip the fraction over and we make the exponent turn positive. Here we want to use the law of exponents. To simplify, we don't want to use any negative exponents, so whenever the bases are the same, we just add the exponents. 2/3 + 1/2 get a common denominator of 6, so 4 sixths plus three sixths, so we get 5 to the seven sixths. This next one when we're dividing, we subtract the exponents, so 7 thirteenths minus a -3 thirteenths minus negatives plus, so we get 9 to the 10 thirteenths. Powers to powers we multiply. So this is five to the. I can't tell on my worksheet. Is that a 3/4? Well, I'm going to pretend it's a 3/4. Might be a 5 fourths. I really just can't tell. Let's say it's a 5 fourths, maybe multiply 5 fourths times 3/10. So the five and the 10 reduced to 2:00. So we get 5 to the 3/8. Remember that may or may not be the original problem on your worksheet. If it's not, please do the one that's on the worksheet powers to powers. We multiply. So A to the -3 halves to the two ninths. We're going to reduce that. We're going to get A to the negative. The twos cancel 3 / 9 gives us 1/3 directions say no negative exponents. So 1 / A to the positive 1/3 powers to powers, and we multiply. So we get X to the negative 112th, Y to the 110th, no negative exponents. So we get Y to the 110th on top, X to the 112th on the bottom. This next one we're going to multiply, so we get M to the three fourteenths, N to the -10 21st. We didn't want any negative exponents, so we'd get M to the three fourteenths over N to the 10:20 first. So now we want to use rational exponents. To simplify, we're not going to use fractional exponents in the final answer. The shortcut method of this is to look and see if the index and the exponent have anything in common. So 6 and four are both divisible by two. So if I thought about 6 / 2 and 4 / 2, that's my final answer. Now the way that many people do it is they put it in fractional form and then they reduce it, and then they put it back into exponential form. But the shortcut is just to see what number divides into both. So 4 and 20 are each divisible by four. 4 / 4 is one, so the Radican is actually going to totally go away. 20 / 4 is five, so that answers just A to the 5th. If we wanted to see it as a fractional, we'd have 20 / 4 and 20 / 4. Is 5-6 and 18 both divisible by 6, so 6 / 6 is one. So the radical portion goes away. 18 / 6 is 3, 7, and 14 sevens in common. 7 / 7 makes it 1, so now we get XY 14 divided by sevens 2, so X ^2 y ^2 here four and two each divisible by two. So 4 / 2 is 2, so we're going to have a square root. 2 / 2 is 1, so we just get sqrt 5 X to the first or just sqrt 5 X Here if I have an index and another index, we actually are just going to multiply this. So four times an understood 2 is going to give me an eighth root. And the reason is that if I think about the fourth root, that square root of X we could think of as X to the half, and then that 4th root we could think of X to the half, to the power of 1/4 powers to powers we multiply. The easy shortcut is just multiply your two indices together. So here 3 * 6 eighteenth root of M here 4:00 and 12:00. One's an index and one's an exponent. So what number divides into both 4 and 12, and that's 4/4 into four goes once. So that Radican goes away XY 12 into 4, four into 12/3. So XY cubed is X ^3, y ^3. Understood. 2 and 16 both divisible by two. So we get AB to the 8th, or A to the 8th and B to the 8th. This last one find the domain of F. Explain how you found your answer. Well, this is really saying we have sqrt X + 5 and because of that negative we have sqrt X + 7. But it's in the denominator and we know that X + 5 has to be greater than or equal to 0 or X has to be greater than or equal to -5. The denominator window can't be equal to 0, so that denominator has to be X + 7 greater than 0 or X greater than -7 I need both of these to be true. So if we thought about it -7 we'd have an open dot shaded off to the right. But at -5 we're going to have a solid dot shaded off to the right. So our solution here is where they're both true. They both have to be true. So our final solution here is going to be -5 to Infinity. If I had thought about using -6 when I put -6 into this numerator, I wouldn't get a real number. Thank you and have a wonderful day.