Surds and Indices: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 1)

Here the base is 2, with exponents 5, 4, and 7. 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 + 4 – 7 = 2. The expression equals 2^2, which computes to 4. Check: 2^2 = 4.

Here the base is 2, with exponents 2, 3, and 4. 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 2 + 3 – 4 = 1. The expression equals 2^1, which computes to 2. Check: 2^1 = 2.

The surd here is the 5-th root of 33, written as 5√33. 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 base is 14, with exponents 11 and 4. The law a^m ÷ a^n = a^(m-n) applies for any base a. So the exponents subtract: 11 – 4 = 7. The expression simplifies to (14)^7.

Here the base is 3, with exponents 7, 4, and 10. 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 7 + 4 – 10 = 1. The expression equals 3^1, which computes to 3. Check: 3^1 = 3.

Here the base is 2, with exponents 9, 4, and 7. 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 9 + 4 – 7 = 6. The expression equals 2^6, which computes to 64. Check: 2^6 = 64.

Here the base is 2, with exponents 7, 10, 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 7 + 10 – 13 = 4. The expression equals 2^4, which computes to 16. Check: 2^4 = 16.

Here the base is 2, with exponents 3, 6, and 2. 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 3 + 6 – 2 = 7. The expression equals 2^7, which computes to 128. Check: 2^7 = 128.

Here the base is 3, with exponents 2, 3, and 3. 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 2 + 3 – 3 = 2. The expression equals 3^2, which computes to 9. Check: 3^2 = 9.

Here the base is 2, with exponents 7, 4, and 5. 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 7 + 4 – 5 = 6. The expression equals 2^6, which computes to 64. Check: 2^6 = 64.

The law (a/b)^n = a^n / b^n holds for any bases a, b (with b nonzero) and exponent n. This follows because raising a quotient to a power means multiplying the quotient by itself n times. Grouping the numerators together and the denominators together gives a^n / b^n. Check with numbers: (6/2)^2 = 3^2 = 9, and 6^2 / 2^2 = 36/4 = 9, which match.

The law (ab)^n = a^n x b^n holds for any bases a, b and exponent n. This follows because raising a product to a power means multiplying the product by itself n times. Grouping the a’s together and the b’s together gives a^n x b^n. Check with numbers: (2×3)^2 = 6^2 = 36, and 2^2 x 3^2 = 4 x 9 = 36, which match.

A surd is a specific kind of irrational number. It is formed by taking a root, such as a square root or cube root, of a rational number. That root must not simplify to a rational number itself. For example, √2 is a surd, since 2 is rational but √2 is irrational. By contrast, √4 = 2 is not a surd, because it simplifies to a rational number.

Evaluate each expression separately. 4¹ = 4, since any number to the power 1 is itself. 1⁴ = 1 x 1 x 1 x 1 = 1. 4⁰ = 1, by the rule that any nonzero number to the power 0 is 1. 0⁴ = 0 x 0 x 0 x 0 = 0. Comparing all four results, only 1⁴ and 4⁰ share the same value, 1.

The law a^0 = 1 holds for any nonzero base a. One way to see why: a^n ÷ a^n = a^(n-n) = a^0, by the subtraction law of indices. But a^n ÷ a^n also equals 1, since any nonzero number divided by itself is 1. So a^0 must equal 1.

Here √3 x √3 needs simplifying, and 3 and 3 are equal here. The law √a x √b = √(ab) applies for non-negative a and b. So √3 x √3 = √9. Since 9 = 3 x 3 is a perfect square, √9 = 3. Check: 3 x 3 = 9, which equals 9.

Here √2 x √2 needs simplifying, and 2 and 2 are equal here. The law √a x √b = √(ab) applies for non-negative a and b. So √2 x √2 = √4. Since 4 = 2 x 2 is a perfect square, √4 = 2. Check: 2 x 2 = 4, which equals 4.

Here √32 x √32 needs simplifying, and 32 and 32 are equal here. The law √a x √b = √(ab) applies for non-negative a and b. So √32 x √32 = √1024. Since 1024 = 32 x 32 is a perfect square, √1024 = 32. Check: 32 x 32 = 1024, which equals 1024.

Here √18 x √18 needs simplifying, and 18 and 18 are equal here. The law √a x √b = √(ab) applies for non-negative a and b. So √18 x √18 = √324. Since 324 = 18 x 18 is a perfect square, √324 = 18. Check: 18 x 18 = 324, which equals 324.

Here √5 x √5 needs simplifying, and 5 and 5 are equal here. The law √a x √b = √(ab) applies for non-negative a and b. So √5 x √5 = √25. Since 25 = 5 x 5 is a perfect square, √25 = 5. Check: 5 x 5 = 25, which equals 25.

Here √7 x √7 needs simplifying, and 7 and 7 are equal here. The law √a x √b = √(ab) applies for non-negative a and b. So √7 x √7 = √49. Since 49 = 7 x 7 is a perfect square, √49 = 7. Check: 7 x 7 = 49, which equals 49.

The surd here is the 5-th root of 40, written as 5√40. 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.

The surd here is the 5-th root of 24, written as 5√24. 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.

The surd here is the 4-th root of 45, written as 4√45. By definition, the order of a surd is the index written on the root symbol. That index is 4 in this case. So the order of the surd is 4.

The surd here is the 6-th root of 19, written as 6√19. By definition, the order of a surd is the index written on the root symbol. That index is 6 in this case. So the order of the surd is 6.