An exponent is repeated multiplication, but exponent questions often test the law used rather than the size of the number. A square root reverses a square; a cube root reverses a cube. The safest approach is to identify the base, preserve brackets and prime-factorise when a root is not immediately visible.
1. Exponent laws and their conditions
For the same non-zero base a, a^m × a^n = a^(m+n), while a^m ÷ a^n = a^(m−n). A power raised to another power multiplies exponents: (a^m)^n = a^(mn). These statements apply to the same base. In general, 2^3 + 2^4 is 8 + 16 = 24, not 2^7. Similarly, (a+b)^2 contains the cross term 2ab; it is not a²+b².
A zero exponent gives a^0 = 1 for a ≠ 0. A negative exponent takes a reciprocal: a^(−n) = 1/a^n. Fractional exponents connect powers to roots: a^(1/2) is the principal square root of non-negative a, while a^(1/3) is its cube root. For even roots in ordinary real arithmetic, the radicand cannot be negative.
Worked example 1. Simplify 3^7 × 3^(−2) ÷ 3^3. The net exponent is 7 − 2 − 3 = 2, so the result is 3² = 9. Compute the exponent first; do not evaluate 3^7 as 2,187 unless needed.
2. Squares and square roots
A perfect square has even exponents in its prime factorisation. For example, 360 = 2³ × 3² × 5. Multiplying by 2 × 5 = 10 makes every exponent even: 3600 = 60². Dividing by 2 × 5 also works when the task asks for the least divisor that leaves a square: 360 ÷ 10 = 36. The words “multiply by” and “divide by” point to different operations even when the prime adjustment happens to be the same.
The square root symbol denotes the non-negative principal root. Thus √49 = 7. The equation x² = 49 has two solutions, x = 7 and x = −7. Confusing these two statements creates unnecessary extra answers. To estimate √50, note that 7² = 49 and 8² = 64; the root is just above 7. Estimation is useful when answer options are far apart.
Worked example 2. Find the smallest positive integer by which 540 must be multiplied to become a perfect square. Since 540 = 2² × 3³ × 5, the odd exponents belong to 3 and 5. Multiply by 15. Then 540 × 15 = 8,100 = 90².
3. Cubes and cube roots
For a perfect cube, each prime exponent is a multiple of three. The number 72 = 2³ × 3² needs one more factor of 3, so 72 × 3 = 216 = 6³. The cube root of a negative number is negative: (−4)³ = −64, hence the real cube root of −64 is −4. This differs from an even root, which cannot produce a negative radicand from a real number.
Memorising small cubes saves time: 1³ through 10³ are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000. Use factorisation or nearby cubes for larger values. Always check the last digit and rough magnitude before settling on a root.
Worked example 3. What is the least divisor that turns 1,728 into a perfect square? Factorise: 1,728 = 2^6 × 3³. The exponent of 3 is odd, so divide by 3. The result is 576 = 24². A perfect cube test would be different: both exponents 6 and 3 are already multiples of three, and 1,728 = 12³.
4. Surds and exact forms
A surd is an irrational root kept in exact form, such as √2. Simplify by taking perfect-square factors outside the radical: √72 = √(36 × 2) = 6√2. Unlike terms can be added only when their remaining radicals agree: 3√2 + 5√2 = 8√2, but 3√2 + 5√3 cannot be combined into 8√5.
For positive numbers, √a × √b = √(ab). The identity needs care when negative radicands are involved, which are outside the real-number treatment used here. Rationalising a simple denominator uses a matching radical: 1/√5 = √5/5. For a denominator a+b√c, multiply top and bottom by the conjugate a−b√c; the denominator becomes a²−b²c.
Worked example 4. Simplify √48 + √75 − √27. The terms become 4√3 + 5√3 − 3√3 = 6√3. The roots were simplified before the coefficients were combined.
5. Unit digits of powers
Only the base's last digit matters for the last digit of an integer power. Powers of 7 cycle through 7, 9, 3 and 1. Reduce the exponent modulo the cycle length. For 7^2026, 2026 leaves remainder 2 on division by 4, so the final digit is the second value, 9. A remainder of zero means the last value in the cycle, not the zeroth value.
The last digits 0, 1, 5 and 6 stay fixed under positive integer powers. The digits 4 and 9 have two-term cycles. Other nontrivial last digits usually have four-term cycles. A calculator is unnecessary once the cycle is known.
Worked example 5. Find the last digit of 3^51 + 8^42. Powers of 3 cycle 3,9,7,1; 51 mod 4 = 3, so 3^51 ends in 7. Powers of 8 cycle 8,4,2,6; 42 mod 4 = 2, so 8^42 ends in 4. The sum ends in 7+4 = 11, hence digit 1.
Recall before practice
1. State why exponents add for multiplication of equal bases but not for addition. 2. Decide whether √81 and the solutions of x²=81 have the same number of values. 3. Use prime exponents to find the least multiplier making 72 a cube. 4. Explain what an exponent-cycle remainder of zero means.
Chapter 4 practice — 50 questions
Choose one option for each item. Keep a separate answer list; explanations follow this set.
1. Simplify: √(0.16 × 0.25)
A. 0.9 B. 0.02 C. 0.2 D. 2
2. Simplify: √6 × √10
A. √60 B. 2√16 C. 3√15 D. 2√15
3. Simplify: 2^1 × 2^6 ÷ 2^3
A. 16 B. 1/16 C. 8 D. 1024
4. If a = √5 + 2, then the value of √(a² − 2√5·a + 5) is:
A. 2 B. 5 C. 4 D. 3
5. Simplify: √(2.25 × 0.64)
A. 0.12 B. 1.2 C. 12 D. 2.3
6. Find the value of √(1 + 3 + 5 + … + 87).
A. 87 B. 43 C. 45 D. 44
7. Rationalise the denominator: 1/(√2 − 1)
A. 1 + √2 B. 1 − √2 C. √2 − 1 D. −1 − √2
8. If 3^? = 9 × 729 × 9, then the value of ? is:
A. -10 B. 24 C. 10 D. 11
9. Simplify: √0.0081 + √0.81 + √81
A. 9.99 B. 99.9 C. 27 D. 0.999
10. Simplify: ∛0.000064 + ∛0.064 + ∛64
A. 0.444 B. 12 C. 44.4 D. 4.44
11. What is the least number that must be subtracted from 1047 to make it a perfect square?
A. 23 B. 32 C. 42 D. 33
12. The value of (1/10)^-2 + (1/6)^-2 + (1/2)^-2 is:
A. 36 B. 18 C. 141 D. 140
13. What is the least number that must be subtracted from 730 to make it a perfect square?
A. 54 B. 27 C. 1 D. 28
14. Simplify: 2^(-3) × 2^1 ÷ 2^1
A. 1/8 B. 8 C. 1/16 D. 1/2
15. Simplify: √(8 + √60)
A. √6 + √2 B. √8 + √15 C. √5 − √3 D. √5 + √3
16. Simplify: √(15 + √104)
A. √13 − √2 B. √14 + 1 C. √13 + √2 D. √15 + √26
17. If √(1 + x/400) = 21/20, then the value of x is:
A. 1 B. 20 C. 441 D. 41
18. If 3^? = 3 × 3 × 243 × 81, then the value of ? is:
A. 12 B. 11 C. 20 D. -11
19. If x = (√26 + 5)/(√26 − 5) and y = (√26 − 5)/(√26 + 5), then the value of x + y is:
A. 102 B. 51 C. 204 D. 10402
20. Find the value of √[(1 + 3 + 5 + … + 35) ÷ (1 + 3 + 5 + … + 5)].
A. 15 B. 18 C. 36 D. 6
21. Simplify: (√3 + √2)/(√3 − √2)
A. 5 + 2√6 B. 2√6 − 5 C. 4 + 2√6 D. 5 − 2√6
22. Find the value of √(74 × 76 + 1).
A. 74 B. 75 C. 76 D. 73
23. If 3^(x + y) = 243 and 7^(3x − y) = 823543, then the value of 8x − 8y is:
A. -16 B. 40 C. 8 D. -8
24. Which of the following is the greatest? √6 + √22, √2 + √26, √11 + √17
A. √6 + √22 B. √2 + √26 C. √11 + √17 D. All are equal
25. Find the value of √[(1 + 3 + 5 + … + 129) ÷ (1 + 3 + 5 + … + 25)].
A. 25 B. 65 C. 52 D. 5
26. Simplify: (√13 + √11)/(√13 − √11)
A. √143 − 12 B. 11 + √143 C. 12 − √143 D. 12 + √143
27. If x = √(72 + √(72 + √(72 + …))), then the value of x is:
A. 10 B. 11 C. 9 D. 8
28. Simplify √24 × √12 to its simplest form.
A. 12√4 B. 12√2 C. √288 D. 24√2
29. If x = √(6 + √(6 + √(6 + …))), then the value of x is:
A. 3 B. 4 C. 2 D. 5
30. If x = 3 − 2√2, then the value of x + 1/x is:
A. 8 B. 4 C. 6 D. 3
31. If √1521 = 39, find the value of √15.21 + √0.1521.
A. 4.29 B. 0.429 C. 42.9 D. 39.39
32. If 2^? = 16 × 64 × 512 × 512, then the value of ? is:
A. 29 B. 28 C. 1944 D. -28
33. Find the least number by which 4704 must be divided so that the quotient is a perfect square.
A. 36 B. 6 C. 28 D. 12
34. Simplify: √(16 + √220)
A. √16 + √55 B. √11 − √5 C. √11 + √5 D. √12 + 2
35. If x = (√5 + 2)/(√5 − 2) and y = (√5 − 2)/(√5 + 2), then the value of x + y is:
A. 9 B. 322 C. 18 D. 36
36. What is the cube root of 85184?
A. 54 B. 44 C. 45 D. 43
37. If x = √(110 + √(110 + √(110 + …))) and y = √(110 − √(110 − √(110 − …))), then x − y is:
A. 3 B. 0 C. 1 D. 2
38. Simplify: (√12 + √11)/(√12 − √11)
A. 4√33 − 23 B. 22 + 4√33 C. 23 − 4√33 D. 23 + 4√33
39. Simplify √24 × √48 to its simplest form.
A. 24√4 B. 24√2 C. 48√2 D. √1152
40. Find the least number by which 392 must be multiplied so that the product is a perfect square.
A. 4 B. 14 C. 28 D. 2
41. If x = 8 + 3√7, then the value of x³ + 1/x³ is:
A. 4096 B. 254 C. 4144 D. 4048
42. Find the value of √(17 + 2√66) − √(17 − 2√66).
A. 2√11 B. 2√6 C. √11 + √6 D. 2√66
43. If 5^(x + y) = 390625 and 2^(3x − y) = 16, then the value of 26x − 33y is:
A. 76 B. 243 C. 31 D. -87
44. If x = (√13 + 3)/(√13 − 3) and y = (√13 − 3)/(√13 + 3), then the value of x² + xy + y² is:
A. 121 B. 120 C. 11 D. 123
45. Find the value of √784 × ∛512.
A. 6272 B. 1792 C. 224 D. 36
46. What is the greatest 3-digit number that is a perfect square?
A. 961 B. 900 C. 1024 D. 999
47. Find the value of √(1 + 2 + 3 + … + 8) + 29.
A. 6 B. 37 C. 65 D. 35
48. Simplify: [2^(x + 7) × 4^(x − 4)] ÷ [8^(x + 0) × 2^5]
A. 1/64 B. 64 C. 16 D. 1/4
49. Which of the following is the smallest? √6 + √18, √10 + √14, √7 + √17, √2 + √22
A. √6 + √18 B. √10 + √14 C. √7 + √17 D. √2 + √22
50. Find the value of √(1 + 2 + 3 + … + 49) + 9.
A. 58 B. 44 C. 35 D. 1234
Chapter 4 — Answers and explanations
After checking the key, re-solve any miss without looking at the formula.
1. C. Write each factor as a perfect square: 0.16 = (0.4)², 0.25 = (0.5)². Hence √(0.16 × 0.25) = 0.4 × 0.5 = 0.2. (Adding the roots gives 0.9, and misplacing the decimal point gives 2 — both wrong.)
APAR26-04-02 | Square root of a product of decimals | Easy
2. D. √6 × √10 = √(6 × 10) = √60 = √(4 × 15) = 2√15. Always take the perfect-square factor out of the radicand.
APAR26-04-27 | Product of surds | Easy
3. A. With the same base, add the exponents when multiplying and subtract when dividing: 2^(1 + 6 − 3) = 2^(4) = 16. (Multiplying the exponents, 1 × 6, is the usual slip.)
APAR26-04-29 | Laws of indices (same base) | Easy
4. A. a² − 2√5·a + 5 = (a − √5)². Its square root is |a − √5| = |√5 + 2 − √5| = 2. (Do not substitute blindly; recognise the perfect square first.)
APAR26-04-26 | Substitution into a perfect-square surd expression | Easy
5. B. Write each factor as a perfect square: 2.25 = (1.5)², 0.64 = (0.8)². Hence √(2.25 × 0.64) = 1.5 × 0.8 = 1.2. (Adding the roots gives 2.3, and misplacing the decimal point gives 12 — both wrong.)
APAR26-04-06 | Square root of a product of decimals | Easy
6. D. The sum of the first n odd numbers is n². Here the last term is 87 = 2n − 1, so n = (87 + 1)/2 = 44 terms, and the sum = 44² = 1936. Therefore √(1936) = 44. (Do not use 87 itself as n.)
APAR26-04-07 | Square root of the sum of odd numbers | Easy
7. A. Multiply numerator and denominator by the conjugate √2 + 1: (√2 + 1)/[(√2)² − 1²] = (√2 + 1)/1. So the value is 1 + √2. (Picking the wrong sign gives √2 − 1.)
APAR26-04-31 | Rationalising the denominator | Easy
8. C. Write each number as a power of 3: 9 = 3^2, 729 = 3^6, 9 = 3^2. Multiplication adds exponents and division subtracts them: ? = 2 + 6 + 2 = 10.
APAR26-04-32 | Find the missing exponent | Easy
9. A. Each term is a perfect square of a decimal: √0.0081 = 0.09, √0.81 = 0.9, √81 = 9 (the number of decimal places halves under the root). Sum = 0.09 + 0.9 + 9 = 9.99. Taking √0.0081 as 0.9 shifts the decimal point wrongly.
APAR26-04-03 | Sum of decimal roots | Easy
10. D. Each term is a perfect cube of a decimal: ∛0.000064 = 0.04, ∛0.064 = 0.4, ∛64 = 4 (the number of decimal places becomes one-third under the root). Sum = 0.04 + 0.4 + 4 = 4.44. Taking ∛0.000064 as 0.4 shifts the decimal point wrongly.
APAR26-04-05 | Sum of decimal roots | Easy
11. A. 32² = 1024 < 1047 < 1089 = 33². The perfect square just below is 1024, so the least number to subtract = 1047 − 1024 = 23. (Adding would need 42.)
APAR26-04-04 | Least number to subtract for a perfect square | Easy
12. D. (1/a)^-n = a^n: invert the fraction and drop the minus sign. So the expression = 10^2 + 6^2 + 2^2 = 100 + 36 + 4 = 140. (Multiplying base by exponent instead of raising is a common slip.)
APAR26-04-28 | Negative exponents of fractions | Easy
13. C. 27² = 729 < 730 < 784 = 28². The perfect square just below is 729, so the least number to subtract = 730 − 729 = 1. (Adding would need 54.)
APAR26-04-01 | Least number to subtract for a perfect square | Easy
14. A. With the same base, add the exponents when multiplying and subtract when dividing: 2^(-3 + 1 − 1) = 2^(-3) = 1/8. (Multiplying the exponents, (−3) × 1, is the usual slip.)
APAR26-04-30 | Laws of indices (same base) | Easy
15. D. Use (√x + √y)² = x + y + 2√(xy). We need x + y = 8 and xy = 15: x = 5, y = 3 work. So √(8 + √60) = √((√5 + √3)²) = √5 + √3. Guessing √8 + √15 is the standard wrong move.
APAR26-04-16 | Square root of a surd expression | Medium
16. C. Use (√x + √y)² = x + y + 2√(xy). We need x + y = 15 and xy = 26: x = 13, y = 2 work. So √(15 + √104) = √((√13 + √2)²) = √13 + √2. Guessing √15 + √26 is the standard wrong move.
APAR26-04-10 | Square root of a surd expression | Medium
17. D. Square both sides: 1 + x/400 = 441/400. So x/400 = (441 − 400)/400 = 41/400, giving x = 41. (Taking x = 21 − 20 = 1 forgets to square.)
APAR26-04-19 | Unknown under a square root | Medium
18. B. Write each number as a power of 3: 3 = 3^1, 3 = 3^1, 243 = 3^5, 81 = 3^4. Multiplication adds exponents and division subtracts them: ? = 1 + 1 + 5 + 4 = 11.
APAR26-04-40 | Find the missing exponent | Medium
19. A. x and y are reciprocals, so xy = 1. x + y = [(√26 + 5)² + (√26 − 5)²]/(26 − 25) = 2(26 + 25)/1 = 102.
APAR26-04-42 | Conjugate surd fractions | Medium
20. D. Sum of the first n odd numbers = n². Numerator: last term 35 ⇒ 18 terms ⇒ 18². Denominator: last term 5 ⇒ 3 terms ⇒ 3². Ratio = 18²/3² = (18/3)², so the square root = 18/3 = 6.
APAR26-04-18 | Ratio of sums of odd numbers under a root | Medium
21. A. Multiply above and below by (√3 + √2): numerator = (√3 + √2)² = 3 + 2 + 2√6 = 5 + 2√6; denominator = 3 − 2 = 1. Result = 5 + 2√6.
APAR26-04-43 | Rationalising a binomial surd fraction | Medium
22. B. Write 74 × 76 = (75 − 1)(75 + 1) = 75² − 1. Adding 1 gives 75², so √(74 × 76 + 1) = √(5625) = 75. The identity (n − 1)(n + 1) + 1 = n² avoids the multiplication 5624 + 1.
APAR26-04-15 | Square root of a(a+2)+1 | Medium
23. C. 243 = 3^5 ⇒ x + y = 5; 823543 = 7^7 ⇒ 3x − y = 7. Adding: 4x = 12 ⇒ x = 3, y = 2. So 8x − 8y = 24 − 16 = 8. (Swapping x and y gives -8.)
APAR26-04-38 | Simultaneous exponential equations | Medium
24. C. Each pair adds to 28, so square them: (√6 + √22)² = 28 + 2√132, (√2 + √26)² = 28 + 2√52, (√11 + √17)² = 28 + 2√187. All share 28; the greatest is the one with the greatest product under the root, 187, i.e. √11 + √17.
APAR26-04-34 | Comparison of surds | Medium
25. D. Sum of the first n odd numbers = n². Numerator: last term 129 ⇒ 65 terms ⇒ 65². Denominator: last term 25 ⇒ 13 terms ⇒ 13². Ratio = 65²/13² = (65/13)², so the square root = 65/13 = 5.
APAR26-04-20 | Ratio of sums of odd numbers under a root | Medium
26. D. Multiply above and below by (√13 + √11): numerator = (√13 + √11)² = 13 + 11 + 2√143 = 24 + 2√143; denominator = 13 − 11 = 2. Result = 12 + √143.
APAR26-04-35 | Rationalising a binomial surd fraction | Medium
27. C. x = √(72 + x) ⇒ x² − x − 72 = 0 ⇒ (x − 9)(x + 8) = 0 ⇒ x = 9 (positive root). Required value = 9. (Shortcut: 72 = 8 × 9; x is the larger factor.)
APAR26-04-39 | Infinite nested surd | Medium
28. B. √24 × √12 = √(24 × 12) = √288. Now 288 = 12² × 2 (2^5 × 3^2), so √288 = 12√2. Leaving the answer as √288 is not the simplest form.
APAR26-04-08 | Product of square roots | Medium
29. A. x = √(6 + x) ⇒ x² − x − 6 = 0 ⇒ (x − 3)(x + 2) = 0 ⇒ x = 3 (positive root). Required value = 3. (Shortcut: 6 = 2 × 3; x is the larger factor.)
APAR26-04-44 | Infinite nested surd | Medium
30. C. 1/x = 1/(3 − 2√2) = (3 + 2√2)/(3² − 8) = 3 + 2√2. So x + 1/x = (3 − 2√2) + (3 + 2√2) = 6; the surd terms cancel.
APAR26-04-45 | x + 1/x for a conjugate surd | Medium
31. A. Moving the decimal point by 2 places under the root moves it by 1 place in the root. √15.21 = 3.9 and √0.1521 = 0.39. Sum = 3.9 + 0.39 = 4.29.
APAR26-04-13 | Shifting the decimal point under a root | Medium
32. B. Write each number as a power of 2: 16 = 2^4, 64 = 2^6, 512 = 2^9, 512 = 2^9. Multiplication adds exponents and division subtracts them: ? = 4 + 6 + 9 + 9 = 28.
APAR26-04-36 | Find the missing exponent | Medium
33. B. Prime factorise: 4704 = 2^5 × 3 × 7^2 = 28² × 6. Every prime must appear an even number of times, so the unpaired part 6 must be removed: 4704 ÷ 6 = 784 = 28². Answer: 6.
APAR26-04-11 | Least number to divide for a perfect square | Medium
34. C. Use (√x + √y)² = x + y + 2√(xy). We need x + y = 16 and xy = 55: x = 11, y = 5 work. So √(16 + √220) = √((√11 + √5)²) = √11 + √5. Guessing √16 + √55 is the standard wrong move.
APAR26-04-17 | Square root of a surd expression | Medium
35. C. x and y are reciprocals, so xy = 1. x + y = [(√5 + 2)² + (√5 − 2)²]/(5 − 4) = 2(5 + 4)/1 = 18.
APAR26-04-33 | Conjugate surd fractions | Medium
36. B. Ignore the last three digits: 85 lies between 4³ = 64 and 5³ = 125, so the tens digit is 4. The unit digit of 85184 is 4, and only 4³ ends in 4, so the unit digit is 4. Cube root = 44. (Prime factorisation: 85184 = 2^6 × 11^3.)
APAR26-04-12 | Cube root of a perfect cube | Medium
37. C. x = √(110 + x) ⇒ x² − x − 110 = 0 ⇒ (x − 11)(x + 10) = 0 ⇒ x = 11 (positive root). Likewise y² = 110 − y ⇒ y² + y − 110 = 0 ⇒ (y + 11)(y − 10) = 0 ⇒ y = 10. Required value = 1. (Shortcut: 110 = 10 × 11; x is the larger factor and y the smaller.)
APAR26-04-41 | Infinite nested surd | Medium
38. D. Multiply above and below by (√12 + √11): numerator = (√12 + √11)² = 12 + 11 + 2√132 = 23 + 4√33; denominator = 12 − 11 = 1. Result = 23 + 4√33.
APAR26-04-37 | Rationalising a binomial surd fraction | Medium
39. B. √24 × √48 = √(24 × 48) = √1152. Now 1152 = 24² × 2 (2^7 × 3^2), so √1152 = 24√2. Leaving the answer as √1152 is not the simplest form.
APAR26-04-14 | Product of square roots | Medium
40. D. Prime factorise: 392 = 2^3 × 7^2 = 14² × 2. Every prime must appear an even number of times, so the unpaired part 2 must be supplied: 392 × 2 = 784 = 28². Answer: 2.
APAR26-04-09 | Least number to multiply for a perfect square | Medium
41. D. Since 8² − 63 = 1, 1/x = 8 − 3√7 and x + 1/x = 16. Then x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x) = 16³ − 3 × 16 = 4096 − 48 = 4048.
APAR26-04-47 | x³ + 1/x³ for a conjugate surd | Difficult
42. B. Since 11 + 6 = 17 and 11 × 6 = 66: √(17 + 2√66) = √11 + √6 and √(17 − 2√66) = √11 − √6. Their difference = (√11 + √6) − (√11 − √6) = 2√6 = 2√6. Their sum would be 2√11.
APAR26-04-23 | Difference of nested roots | Difficult
43. D. 390625 = 5^8 ⇒ x + y = 8; 16 = 2^4 ⇒ 3x − y = 4. Adding: 4x = 12 ⇒ x = 3, y = 5. So 26x − 33y = 78 − 165 = -87. (Swapping x and y gives 31.)
APAR26-04-49 | Simultaneous exponential equations | Difficult
44. B. x and y are reciprocals, so xy = 1. x + y = [(√13 + 3)² + (√13 − 3)²]/(13 − 9) = 2(13 + 9)/4 = 11. Then x² + xy + y² = (x + y)² − xy = 11² − 1 = 120.
APAR26-04-48 | Conjugate surd fractions | Difficult
45. C. √784 = 28 (28² = 784) and ∛512 = 8 (8³ = 512). Product = 28 × 8 = 224. Taking √512 in place of the cube root, or adding instead of multiplying (36), are the common errors.
APAR26-04-21 | Square root and cube root combined | Difficult
46. A. The greatest 3-digit number is 999. √999 ≈ 31.61, so the required root is 31 and the number is 31² = 961. (1024 has 4 digits.)
APAR26-04-24 | greatest 3-digit perfect square | Difficult
47. D. 1 + 2 + … + 8 = 8 × 9/2 = 36 = 6². So the root is 6 and the value = 6 + 29 = 35. (This is not the odd-number sum, so the answer is not 8 + 29.)
APAR26-04-22 | Square root of a triangular sum | Difficult
48. A. Convert to base 2: 4^(x − 4) = 2^(2x − 8) and 8^(x + 0) = 2^(3x). Total exponent = (x + 7) + (2x − 8) − (3x) − 5 = -6; the x-terms cancel (x + 2x − 3x = 0). Value = 2^(-6) = 1/64.
APAR26-04-46 | Exponent expression where x cancels | Difficult
49. D. Each pair adds to 24, so square them: (√6 + √18)² = 24 + 2√108, (√10 + √14)² = 24 + 2√140, (√7 + √17)² = 24 + 2√119, (√2 + √22)² = 24 + 2√44. All share 24; the smallest is the one with the smallest product under the root, 44, i.e. √2 + √22.
APAR26-04-50 | Comparison of surds | Difficult
50. B. 1 + 2 + … + 49 = 49 × 50/2 = 1225 = 35². So the root is 35 and the value = 35 + 9 = 44. (This is not the odd-number sum, so the answer is not 49 + 9.)
APAR26-04-25 | Square root of a triangular sum | Difficult