Answer :
Sure, let's go through each problem step by step.
### Problem 308
[tex]\[ \sqrt{64 \cdot 121} \][/tex]
1. Calculate the product of 64 and 121:
[tex]\[ 64 \cdot 121 = 7744 \][/tex]
2. Take the square root of 7744:
[tex]\[ \sqrt{7744} = 88 \][/tex]
So, [tex]\[ \sqrt{64 \cdot 121} = 88 \][/tex]
### Problem 316
[tex]\[ \sqrt[5]{(-32)(-243)} \][/tex]
1. Calculate the product of -32 and -243:
[tex]\[ (-32) \cdot (-243) = 7776 \][/tex]
2. Take the 5th root of 7776:
[tex]\[ \sqrt[5]{7776} = 6 \][/tex]
So, [tex]\[ \sqrt[5]{(-32)(-243)} = 6 \][/tex]
### Problem 309
[tex]\[ \sqrt[3]{(-125)(343)} \][/tex]
1. Calculate the product of -125 and 343:
[tex]\[ (-125) \cdot 343 = -42875 \][/tex]
2. Find the cube root of -42875:
[tex]\[ \sqrt[3]{-42875} \][/tex]
Since we're dealing with a negative number, the cube root would be complex:
[tex]\[ \sqrt[3]{-42875} \approx 17.5 + 30.310889132455344j \][/tex]
### Problem 317
[tex]\[ \sqrt{729} \div 81 \][/tex]
1. Calculate the square root of 729:
[tex]\[ \sqrt{729} = 27 \][/tex]
2. Divide by 81:
[tex]\[ 27 \div 81 = \frac{1}{3} \][/tex]
So, [tex]\[ \sqrt{729} \div 81 = \frac{1}{3} \][/tex]
### Problem 310
[tex]\[ \sqrt[6]{7^6 \cdot 2^{12}} \][/tex]
1. Calculate [tex]\( 7^6 \)[/tex]:
[tex]\[ 7^6 = 117649 \][/tex]
2. Calculate [tex]\( 2^{12} \)[/tex]:
[tex]\[ 2^{12} = 4096 \][/tex]
3. Multiply the two products:
[tex]\[ 7^6 \cdot 2^{12} = 117649 \cdot 4096 \][/tex]
4. Take the 6th root of the product:
[tex]\[ \sqrt[6]{117649 \cdot 4096} = 28 \][/tex]
So, [tex]\[ \sqrt[6]{7^6 \cdot 2^{12}} = 28 \][/tex]
### Problem 318
[tex]\[ \sqrt[5]{(-3)^{10}(-5)^5} \][/tex]
1. Calculate [tex]\( (-3)^{10} \)[/tex]:
[tex]\[ (-3)^{10} = 59049 \][/tex]
2. Calculate [tex]\( (-5)^5 \)[/tex]:
[tex]\[ (-5)^5 = -3125 \][/tex]
3. Multiply the two products:
[tex]\[ 59049 \cdot -3125 = -184570625 \][/tex]
4. Take the 5th root:
[tex]\[ \sqrt[5]{-184570625} = 45 \][/tex]
So, [tex]\[ \sqrt[5]{(-3)^{10}(-5)^5} = 45 \][/tex]
### Problem 311
[tex]\[ \sqrt[2]{(-8)^{12} \div (-8)^6} \][/tex]
1. Calculate [tex]\( (-8)^{12} \)[/tex]:
[tex]\[ (-8^{12}) = 68719476736 \][/tex]
2. Calculate [tex]\( (-8)^6 \)[/tex]:
[tex]\[ (-8)^6 = 262144 \][/tex]
3. Divide the two results:
[tex]\[ (-8)^{12} \div (-8)^6 = \frac{68719476736}{262144} = 262144 \][/tex]
4. Take the square root:
[tex]\[ \sqrt{262144} = 512 \][/tex]
So, [tex]\[ \sqrt[2]{(-8)^{12} \div (-8)^6} = 512 \][/tex]
### Problem 319
[tex]\[ \sqrt[4]{2^8 \cdot 3^{12} \cdot 4^4} \][/tex]
1. Calculate [tex]\( 2^8 \)[/tex]:
[tex]\[ 2^8 = 256 \][/tex]
2. Calculate [tex]\( 3^{12} \)[/tex]:
[tex]\[ 3^{12} = 531441 \][/tex]
3. Calculate [tex]\( 4^4 \)[/tex]:
[tex]\[ 4^4 = 256 \][/tex]
4. Multiply the results:
[tex]\[ 256 \cdot 531441 \cdot 256 \][/tex]
5. Take the 4th root:
[tex]\[ \sqrt[4]{256 \cdot 531441 \cdot 256} \approx 432 \][/tex]
So, [tex]\[ \sqrt[4]{2^8 \cdot 3^{12} \cdot 4^4} = 432 \][/tex]
### Problem 312
[tex]\[ \sqrt[4]{3^{2^3}} \][/tex]
1. Calculate the exponent [tex]\( 2^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
2. Calculate [tex]\( 3^8 \)[/tex]:
[tex]\[ 6561 \][/tex]
3. Take the 4th root:
[tex]\[ \sqrt[4]{6561} = 9 \][/tex]
So, [tex]\[ \sqrt[4]{3^{2^3}} = 9 \][/tex]
### Problem 320
[tex]\[ \sqrt[25]{4^{5^2}} \][/tex]
1. Calculate the exponent [tex]\( 5^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
2. Calculate [tex]\( 4^{25} \)[/tex]:
[tex]\[ 4^{25} \][/tex]
3. Take the 25th root:
[tex]\[ \sqrt[25]{4^{25}} = 4 \][/tex]
So, [tex]\[ \sqrt[25]{4^{5^2}} = 4 \][/tex]
### Problem 313
[tex]\[ \sqrt[8]{6^{4^2}} \][/tex]
1. Calculate the exponent [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
2. Calculate [tex]\( 6^{16} \)[/tex]:
[tex]\[ 6^{16} \][/tex]
3. Take the 8th root:
[tex]\[ \sqrt[8]{6^{16}} = 36 \][/tex]
So, [tex]\[ \sqrt[8]{6^{4^2}} = 36 \][/tex]
### Problem 321
[tex]\[ \sqrt[32]{12^4} \][/tex]
1. Calculate [tex]\( 12^4 \)[/tex]:
[tex]\[ 12^4 = 20736 \][/tex]
2. Take the 32nd root:
[tex]\[ \sqrt[32]{20736} \approx 1.364 \][/tex]
So, [tex]\[ \sqrt[32]{12^4} \approx 1.364 \][/tex]
### Problem 322
[tex]\[ \sqrt[4]{(-3)^{4^2}} \][/tex]
1. Calculate the exponent [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
2. Calculate [tex]\( (-3)^{16} \)[/tex]:
[tex]\[ (-3)^{16} \][/tex]
3. Take the 4th root:
[tex]\[ \sqrt[4]{(-3)^{16}} = 81 \][/tex]
So, [tex]\[ \sqrt[4]{(-3)^{4^2}} = 81 \][/tex]
### Problem 308
[tex]\[ \sqrt{64 \cdot 121} \][/tex]
1. Calculate the product of 64 and 121:
[tex]\[ 64 \cdot 121 = 7744 \][/tex]
2. Take the square root of 7744:
[tex]\[ \sqrt{7744} = 88 \][/tex]
So, [tex]\[ \sqrt{64 \cdot 121} = 88 \][/tex]
### Problem 316
[tex]\[ \sqrt[5]{(-32)(-243)} \][/tex]
1. Calculate the product of -32 and -243:
[tex]\[ (-32) \cdot (-243) = 7776 \][/tex]
2. Take the 5th root of 7776:
[tex]\[ \sqrt[5]{7776} = 6 \][/tex]
So, [tex]\[ \sqrt[5]{(-32)(-243)} = 6 \][/tex]
### Problem 309
[tex]\[ \sqrt[3]{(-125)(343)} \][/tex]
1. Calculate the product of -125 and 343:
[tex]\[ (-125) \cdot 343 = -42875 \][/tex]
2. Find the cube root of -42875:
[tex]\[ \sqrt[3]{-42875} \][/tex]
Since we're dealing with a negative number, the cube root would be complex:
[tex]\[ \sqrt[3]{-42875} \approx 17.5 + 30.310889132455344j \][/tex]
### Problem 317
[tex]\[ \sqrt{729} \div 81 \][/tex]
1. Calculate the square root of 729:
[tex]\[ \sqrt{729} = 27 \][/tex]
2. Divide by 81:
[tex]\[ 27 \div 81 = \frac{1}{3} \][/tex]
So, [tex]\[ \sqrt{729} \div 81 = \frac{1}{3} \][/tex]
### Problem 310
[tex]\[ \sqrt[6]{7^6 \cdot 2^{12}} \][/tex]
1. Calculate [tex]\( 7^6 \)[/tex]:
[tex]\[ 7^6 = 117649 \][/tex]
2. Calculate [tex]\( 2^{12} \)[/tex]:
[tex]\[ 2^{12} = 4096 \][/tex]
3. Multiply the two products:
[tex]\[ 7^6 \cdot 2^{12} = 117649 \cdot 4096 \][/tex]
4. Take the 6th root of the product:
[tex]\[ \sqrt[6]{117649 \cdot 4096} = 28 \][/tex]
So, [tex]\[ \sqrt[6]{7^6 \cdot 2^{12}} = 28 \][/tex]
### Problem 318
[tex]\[ \sqrt[5]{(-3)^{10}(-5)^5} \][/tex]
1. Calculate [tex]\( (-3)^{10} \)[/tex]:
[tex]\[ (-3)^{10} = 59049 \][/tex]
2. Calculate [tex]\( (-5)^5 \)[/tex]:
[tex]\[ (-5)^5 = -3125 \][/tex]
3. Multiply the two products:
[tex]\[ 59049 \cdot -3125 = -184570625 \][/tex]
4. Take the 5th root:
[tex]\[ \sqrt[5]{-184570625} = 45 \][/tex]
So, [tex]\[ \sqrt[5]{(-3)^{10}(-5)^5} = 45 \][/tex]
### Problem 311
[tex]\[ \sqrt[2]{(-8)^{12} \div (-8)^6} \][/tex]
1. Calculate [tex]\( (-8)^{12} \)[/tex]:
[tex]\[ (-8^{12}) = 68719476736 \][/tex]
2. Calculate [tex]\( (-8)^6 \)[/tex]:
[tex]\[ (-8)^6 = 262144 \][/tex]
3. Divide the two results:
[tex]\[ (-8)^{12} \div (-8)^6 = \frac{68719476736}{262144} = 262144 \][/tex]
4. Take the square root:
[tex]\[ \sqrt{262144} = 512 \][/tex]
So, [tex]\[ \sqrt[2]{(-8)^{12} \div (-8)^6} = 512 \][/tex]
### Problem 319
[tex]\[ \sqrt[4]{2^8 \cdot 3^{12} \cdot 4^4} \][/tex]
1. Calculate [tex]\( 2^8 \)[/tex]:
[tex]\[ 2^8 = 256 \][/tex]
2. Calculate [tex]\( 3^{12} \)[/tex]:
[tex]\[ 3^{12} = 531441 \][/tex]
3. Calculate [tex]\( 4^4 \)[/tex]:
[tex]\[ 4^4 = 256 \][/tex]
4. Multiply the results:
[tex]\[ 256 \cdot 531441 \cdot 256 \][/tex]
5. Take the 4th root:
[tex]\[ \sqrt[4]{256 \cdot 531441 \cdot 256} \approx 432 \][/tex]
So, [tex]\[ \sqrt[4]{2^8 \cdot 3^{12} \cdot 4^4} = 432 \][/tex]
### Problem 312
[tex]\[ \sqrt[4]{3^{2^3}} \][/tex]
1. Calculate the exponent [tex]\( 2^3 \)[/tex]:
[tex]\[ 2^3 = 8 \][/tex]
2. Calculate [tex]\( 3^8 \)[/tex]:
[tex]\[ 6561 \][/tex]
3. Take the 4th root:
[tex]\[ \sqrt[4]{6561} = 9 \][/tex]
So, [tex]\[ \sqrt[4]{3^{2^3}} = 9 \][/tex]
### Problem 320
[tex]\[ \sqrt[25]{4^{5^2}} \][/tex]
1. Calculate the exponent [tex]\( 5^2 \)[/tex]:
[tex]\[ 5^2 = 25 \][/tex]
2. Calculate [tex]\( 4^{25} \)[/tex]:
[tex]\[ 4^{25} \][/tex]
3. Take the 25th root:
[tex]\[ \sqrt[25]{4^{25}} = 4 \][/tex]
So, [tex]\[ \sqrt[25]{4^{5^2}} = 4 \][/tex]
### Problem 313
[tex]\[ \sqrt[8]{6^{4^2}} \][/tex]
1. Calculate the exponent [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
2. Calculate [tex]\( 6^{16} \)[/tex]:
[tex]\[ 6^{16} \][/tex]
3. Take the 8th root:
[tex]\[ \sqrt[8]{6^{16}} = 36 \][/tex]
So, [tex]\[ \sqrt[8]{6^{4^2}} = 36 \][/tex]
### Problem 321
[tex]\[ \sqrt[32]{12^4} \][/tex]
1. Calculate [tex]\( 12^4 \)[/tex]:
[tex]\[ 12^4 = 20736 \][/tex]
2. Take the 32nd root:
[tex]\[ \sqrt[32]{20736} \approx 1.364 \][/tex]
So, [tex]\[ \sqrt[32]{12^4} \approx 1.364 \][/tex]
### Problem 322
[tex]\[ \sqrt[4]{(-3)^{4^2}} \][/tex]
1. Calculate the exponent [tex]\( 4^2 \)[/tex]:
[tex]\[ 4^2 = 16 \][/tex]
2. Calculate [tex]\( (-3)^{16} \)[/tex]:
[tex]\[ (-3)^{16} \][/tex]
3. Take the 4th root:
[tex]\[ \sqrt[4]{(-3)^{16}} = 81 \][/tex]
So, [tex]\[ \sqrt[4]{(-3)^{4^2}} = 81 \][/tex]
