Rewrite evaluate the followings
1. 32^4/5
2. 125^2/3
3. 1296^3/4
4. 243^4/5
5. 8^2/3
6. 27^2/3
7. 16^3/2
8. 625^5/4
32 = 2^5, so 2 = 32^1/5
so, 32^4/5 = 2^4 = 16
The others are all similar. Find the root using the fraction's denominator, then raise it to the power given in the numerator.
To evaluate the given expressions, we can follow the basic rules of exponentiation.
1. 32^4/5:
To solve this expression, we need to take the fourth power of 32 and then take the fifth root of the result.
Step 1: 32^4 = 32 * 32 * 32 * 32 = 1,048,576
Step 2: Take the fifth root of 1,048,576, which gives us approximately 28.18.
2. 125^2/3:
Similarly, for this expression, we first take the square of 125 and then take the cube root of the result.
Step 1: 125^2 = 125 * 125 = 15,625
Step 2: Take the cube root of 15,625, which gives us the exact value of 25.
3. 1296^3/4:
For this expression, we need to cube 1296 and then take the fourth root of the result.
Step 1: 1296^3 = 1,679,616,256
Step 2: Take the fourth root of 1,679,616,256, which gives us the exact value of 144.
4. 243^4/5:
In this case, we raise 243 to the power of four and then take the fifth root of the result.
Step 1: 243^4 = 243 * 243 * 243 * 243 = 2,781,218,943
Step 2: Take the fifth root of 2,781,218,943, which gives us approximately 20.76.
5. 8^2/3:
Following the same process, we raise 8 to the power of two and then take the cube root of the result.
Step 1: 8^2 = 8 * 8 = 64
Step 2: Take the cube root of 64, which gives us the exact value of 4.
6. 27^2/3:
Again, we raise 27 to the power of two and then take the cube root of the result.
Step 1: 27^2 = 27 * 27 = 729
Step 2: Take the cube root of 729, which gives us the exact value of 9.
7. 16^3/2:
In this case, we cube 16 and then take the square root of the result.
Step 1: 16^3 = 16 * 16 * 16 = 4096
Step 2: Take the square root of 4096, which gives us the exact value of 64.
8. 625^5/4:
For this expression, we raise 625 to the power of five and then take the fourth root of the result.
Step 1: 625^5 = 625 * 625 * 625 * 625 * 625 = 9,765,625,000,000
Step 2: Take the fourth root of 9,765,625,000,000, which gives us approximately 6,250.