Exponents

Correct answer: C. 533

Explanation: Evaluate the powers first: \(23^2=529\) and \(2^2=4\). Therefore, \(23^2+2^2=529+4=533\), so option C is correct. A value such as 523 can result from an addition error. Exam tip: calculate each exponent before performing the addition.

Correct answer: D. 500

Explanation: First evaluate the exponent: \(24^2=24\times24=576\). Then \(576-76=500\). Therefore, the correct answer is 500. Option 510 may result from an error in subtraction. Exam tip: In an expression with brackets, evaluate exponents before subtraction.

Correct answer: A. \(a^m\times a^n\)

Explanation: When powers with the same base are multiplied, their exponents are added: \(a^m\times a^n=a^{m+n}\). In contrast, \(a^m\div a^n=a^{m-n}\), while \((a^m)^n=a^{mn}\). Exam tip: add exponents for multiplication, subtract for division, and multiply them for a power of a power.

Correct answer: C. 7

Explanation: Since \(343=7\times7\times7=7^3\), the equation \(c^3=343\) gives \(c=7\). For example, \(6^3=216\), so 6 is not correct. Exam tip: To find a cube root, identify the number which, when multiplied by itself three times, gives the given number.

Correct answer: D. 576

Explanation: First find the square of 26: \(26^2=26\times26=676\). Then \(676-100=576\), so option D is correct. The value 566 would result from subtracting 110 from 676, so it is not correct. Exam tip: Evaluate exponents first, then perform addition or subtraction.

Correct answer: A. 6

Explanation: When powers with the same base are divided, subtract the exponents: \(6^3\div6^2=6^{3-2}=6^1=6\). Option 36 is \(6^2\), which may result from applying the exponent rule incorrectly. Exam tip: use \(a^m\div a^n=a^{m-n}\) for the same non-zero base.

Correct answer: C. 240

Explanation: Here, \(2^8=256\) and \(2^4=16\). Therefore, \(2^8-2^4=256-16=240\), so 240 is the correct option. An answer such as 250 can result from an error in subtraction. Exam tip: evaluate each power separately before subtracting.

Correct answer: D. 500

Explanation: First find the square of 28: \(28^2=28\times28=784\). Then \(784-284=500\). Therefore, the correct answer is 500. Getting 490 may result from an error in subtraction. Exam tip: evaluate the exponent first, then perform the remaining operations.

Correct answer: C. 1000

Explanation: Here, 30^2 means 30 × 30, which equals 900. Therefore, 30^2 + 100 = 900 + 100 = 1000. Option A includes only the value of 30^2 and does not add 100. Exam tip: Evaluate the exponent first, then perform addition or subtraction.

Correct answer: D. 600

Explanation: First evaluate the exponent: \(31^2=31\times31=961\). Then \(961-361=600\), so 600 is the correct option. A value such as 610 may result from an error in subtraction. Exam tip: In expressions with exponents, evaluate the exponent before performing addition or subtraction.

Correct answer: C. 648

Explanation: Compute the powers first: \(9^3=729\) and \(9^2=81\). So \(9^3-9^2=729-81=648\). Alternate method: factor out \(9^2\): \(9^2(9-1)=81\times8=648\). The closest distractor 658 is a simple arithmetic slip (off by 10); 729 would be the result of forgetting to subtract, and 81 is just \(9^2\). Exam tip: factor the common power \(a^n(a-1)\) or compute powers then subtract to avoid mistakes.

Correct answer: B. 600

Explanation: First evaluate the exponent: \(33^2 = 33 \times 33 = 1089\). Then \(1089 - 489 = 600\). Therefore, the correct answer is 600. The value 1089 is only \(33^2\); the subtraction of 489 must still be performed. Exam tip: In a mixed expression, evaluate exponents before addition or subtraction.

Correct answer: A. 18

Explanation: Since the square root of 324 is 18, the possible values of d are 18 and -18. As the question asks for the positive value, d = 18 is correct. Option 16 is not correct because 16² = 256, not 324. Exam tip: When the positive value is asked in d² = n, take the positive square root of n.

Correct answer: C. 1250

Explanation: First find the square of 35: 35² = 1225. Adding 25 gives 1225 + 25 = 1250. Therefore, 1250 is the correct answer. Option 1225 is only the value of 35²; it does not include the added 25. Exam tip: evaluate the exponent first, then perform addition or subtraction.

Correct answer: D. 600

Explanation: First find the square of 36: \(36^2=36\times36=1296\). Then \(1296-696=600\), so the correct answer is 600. The option 650 can result from an error in subtraction. Exam tip: after finding a square, verify the subtraction using column subtraction or a quick mental check.

Correct answer: B. 91

Explanation: Evaluate the powers first: \(2^6=64\) and \(3^3=27\). Therefore, \(2^6+3^3=64+27=91\). Hence, 91 is the correct answer. An option such as 81 does not result from correctly evaluating and adding both powers. Exam tip: In such expressions, calculate exponents before performing addition or subtraction.

Correct answer: C. 8

Explanation: Given x^3=512. Since 8^3=8×8×8=512, x=8. Option 7 is not correct because 7^3=343. Exam tip: Memorising cubes of numbers from 1 to 10 helps solve such questions quickly.

Correct answer: D. 1000

Explanation: First find the square of 38: \(38^2=38\times38=1444\). Then, \(1444-444=1000\), so the correct answer is 1000. An answer such as 900 can result from an error in finding the square or in subtraction. Exam tip: In an expression with brackets, evaluate the exponent (square) before performing subtraction.

Correct answer: A. 49

Explanation: When powers with the same base are divided, subtract the exponents: \(7^4 \div 7^2 = 7^{4-2} = 7^2 = 49\). Therefore, 49 is correct. 343 is \(7^3\), so it does not result here. Exam tip: use \(a^m \div a^n = a^{m-n}\) only when the non-zero bases are the same.

Correct answer: D. 1000

Explanation: First evaluate the exponent: \(40^2 = 40 \times 40 = 1600\). Then \(1600 - 600 = 1000\). Therefore, the correct answer is 1000. An option such as 1100 may result from an error in subtraction. Exam tip: In an expression with brackets, calculate the exponent before performing subtraction.

Correct answer: A. 21

Explanation: Since 21 × 21 = 441, the positive square root of 441 is 21. Although the equation y² = 441 has two solutions, 21 and −21, the question asks for the positive value, so 21 is correct. The close distractor 19 is incorrect because 19² = 361. Exam tip: when the positive value is asked in y² = a, take √a.

Correct answer: A. 1008

Explanation: Compute powers: \(4^5=1024\) and \(4^2=16\), so \(1024-16=1008\). Why other options are wrong: 1024 corresponds to forgetting the subtraction step; 16 is just \(4^2\) alone; 1000 is a rounded estimate. Exam tip: factor out \(4^2\): \(4^5-4^2=4^2(4^3-1)=16(64-1)=16\times63=1008\), which helps avoid large subtraction errors.

Correct answer: D. 1000

Explanation: First find the square of 42: 42² = 1764. Subtracting 764 from 1764 gives 1000. Therefore, the correct answer is 1000. An option such as 1100 may result from an error in subtraction at the hundreds place. Exam tip: In an expression with exponents, evaluate the exponent first and then perform addition or subtraction.

Correct answer: A. 81

Explanation: When powers with the same base are divided, subtract their exponents: \(9^4 \div 9^2 = 9^{4-2}=9^2=81\). Therefore, 81 is correct. The option 9 is only \(9^1\), so it is not correct. Exam tip: use \(a^m \div a^n=a^{m-n}\), where \(a\ne0\).

Correct answer: C. 9

Explanation: Since 9 × 9 × 9 = 729, we get z = 9. The cube of 8 is 512, so 8 is not correct. Exam tip: verify a perfect cube by multiplying the number by itself three times.