Surds and Indices: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 2)

The surd here is the 5-th root of 34, written as 5√34. By definition, the order of a surd is the index written on the root symbol. That index is 5 in this case. So the order of the surd is 5.

Here the equation is (4)^y = 16. Both 4 and 16 share the same prime base: 4 = 2^2 and 16 = 2^4. Substituting gives (2^2)^y = 2^4. By the law (a^m)^n = a^(mn), the left side becomes 2^(2y). Matching exponents of equal bases: 2y = 4. So y = 4 ÷ 2 = 2. Check: (4)^2 = 16.

Here the equation is (16)^y = 16. Both 16 and 16 share the same prime base: 16 = 2^4 and 16 = 2^4. Substituting gives (2^4)^y = 2^4. By the law (a^m)^n = a^(mn), the left side becomes 2^(4y). Matching exponents of equal bases: 4y = 4. So y = 4 ÷ 4 = 1. Check: (16)^1 = 16.

Here the equation is (625)^y = 625. Both 625 and 625 share the same prime base: 625 = 5^4 and 625 = 5^4. Substituting gives (5^4)^y = 5^4. By the law (a^m)^n = a^(mn), the left side becomes 5^(4y). Matching exponents of equal bases: 4y = 4. So y = 4 ÷ 4 = 1. Check: (625)^1 = 625.

Here the equation is (81)^y = 81. Both 81 and 81 share the same prime base: 81 = 3^4 and 81 = 3^4. Substituting gives (3^4)^y = 3^4. By the law (a^m)^n = a^(mn), the left side becomes 3^(4y). Matching exponents of equal bases: 4y = 4. So y = 4 ÷ 4 = 1. Check: (81)^1 = 81.

Here the equation is (9)^y = 9. Both 9 and 9 share the same prime base: 9 = 3^2 and 9 = 3^2. Substituting gives (3^2)^y = 3^2. By the law (a^m)^n = a^(mn), the left side becomes 3^(2y). Matching exponents of equal bases: 2y = 2. So y = 2 ÷ 2 = 1. Check: (9)^1 = 9.

Here (4)^x = 16, and we need to find x. Writing 16 as 4 multiplied by itself repeatedly shows 16 = 4^2. Since the bases on both sides now match, the exponents must be equal. So x = 2. Check: 4^2 = 16, which matches the given equation.

Here (3)^x = 729, and we need to find x. Writing 729 as 3 multiplied by itself repeatedly shows 729 = 3^6. Since the bases on both sides now match, the exponents must be equal. So x = 6. Check: 3^6 = 729, which matches the given equation.

Here (4)^x = 262144, and we need to find x. Writing 262144 as 4 multiplied by itself repeatedly shows 262144 = 4^9. Since the bases on both sides now match, the exponents must be equal. So x = 9. Check: 4^9 = 262144, which matches the given equation.

Here (8)^x = 512, and we need to find x. Writing 512 as 8 multiplied by itself repeatedly shows 512 = 8^3. Since the bases on both sides now match, the exponents must be equal. So x = 3. Check: 8^3 = 512, which matches the given equation.

Here (3)^x = 243, and we need to find x. Writing 243 as 3 multiplied by itself repeatedly shows 243 = 3^5. Since the bases on both sides now match, the exponents must be equal. So x = 5. Check: 3^5 = 243, which matches the given equation.

Here (7)^x = 2401, and we need to find x. Writing 2401 as 7 multiplied by itself repeatedly shows 2401 = 7^4. Since the bases on both sides now match, the exponents must be equal. So x = 4. Check: 7^4 = 2401, which matches the given equation.

Here (8)^x = 64, and we need to find x. Writing 64 as 8 multiplied by itself repeatedly shows 64 = 8^2. Since the bases on both sides now match, the exponents must be equal. So x = 2. Check: 8^2 = 64, which matches the given equation.

Here (6)^x = 7776, and we need to find x. Writing 7776 as 6 multiplied by itself repeatedly shows 7776 = 6^5. Since the bases on both sides now match, the exponents must be equal. So x = 5. Check: 6^5 = 7776, which matches the given equation.

Here (3)^x = 2187, and we need to find x. Writing 2187 as 3 multiplied by itself repeatedly shows 2187 = 3^7. Since the bases on both sides now match, the exponents must be equal. So x = 7. Check: 3^7 = 2187, which matches the given equation.

Here (10)^x = 1000000, and we need to find x. Writing 1000000 as 10 multiplied by itself repeatedly shows 1000000 = 10^6. Since the bases on both sides now match, the exponents must be equal. So x = 6. Check: 10^6 = 1000000, which matches the given equation.

Here the base is 16, with exponents 17, 16, and 13. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 17 + 16 – 13 = 20. The expression simplifies to (16)^20.

Here the base is 10, with exponents 10, 12, and 20. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 10 + 12 – 20 = 2. The expression simplifies to (10)^2.

Here the base is 10, with exponents 13, 12, and 24. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 13 + 12 – 24 = 1. The expression simplifies to (10)^1.

Here the base is 14, with exponents 18, 19, and 16. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 18 + 19 – 16 = 21. The expression simplifies to (14)^21.

Here the base is 16, with exponents 19, 20, and 22. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 19 + 20 – 22 = 17. The expression simplifies to (16)^17.

Here the base is 19, with exponents 12, 12, and 15. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 12 + 12 – 15 = 9. The expression simplifies to (19)^9.

Here the base is 20, with exponents 11, 11, and 18. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 11 + 11 – 18 = 4. The expression simplifies to (20)^4.

Here the base is 14, with exponents 12, 5, and 15. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 12 + 5 – 15 = 2. The expression simplifies to (14)^2.

Here the base is 3, with exponents 5, 12, and 14. Multiplying keeps the same base and adds exponents: a^m x a^n = a^(m+n). Dividing then subtracts the next exponent: a^(m+n) ÷ a^p = a^(m+n-p). So the exponent is 5 + 12 – 14 = 3. The expression simplifies to (3)^3.