Exponents

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((3^3)^2=3^{3\times2}=3^6=729\). Option 243 equals \(3^5\), so it is not correct. Exam tip: when a power is raised to another power, multiply the exponents.

Since \(125=5^3\), the equation \(a^3=125\) gives \(a=5\). Option 4 is incorrect because \(4^3=64\), not 125. Exam tip: remember common perfect cubes such as \(3^3=27\), \(4^3=64\), and \(5^3=125\).

Evaluate the powers first: \(2^5=32\) and \(2^4=16\). Therefore, \(2^5+2^4=32+16=48\), so 48 is correct. The value 32 represents only \(2^5\); \(2^4\) still has to be added. Exam tip: In expressions with exponents, evaluate the powers before performing addition or subtraction.

Evaluate the powers first: \(3^4=81\) and \(3^2=9\). Therefore, \(3^4-3^2=81-9=72\). Option 81 is only the value of \(3^4\), not the result after subtraction. Exam tip: in expressions with mixed operations, evaluate exponents before addition or subtraction.

Since \(4 = 4^1\), subtract the exponents when dividing powers with the same base: \(4^3 \div 4^1 = 4^{3-1} = 4^2 = 16\). Therefore, 16 is correct. The value 64 is \(4^3\) before dividing by 4. Exam tip: use \(a^m \div a^n = a^{m-n}\) for powers with the same base.

When powers with the same base are multiplied, their exponents are added: \(5^2 \times 5^2 = 5^{2+2}=5^4=625\). Therefore, 625 is the correct answer. 25 is only the value of \(5^2\), while 3125 equals \(5^5\). Exam tip: for multiplication of the same base, add the exponents.

Any non-zero number raised to the power zero equals 1, so \(6^0=1\). Therefore, \(6^0+1=1+1=2\). Option 1 is only the value of \(6^0\); the given \(+1\) must also be added. Exam tip: remember that \(a^0=1\) for every non-zero base \(a\).

Since \(12^2=144\), the solutions of \(x^2=144\) are \(x=12\) and \(x=-12\). The question asks for the positive value, so the answer is \(12\). Since \(14^2=196\), 14 is not correct. Exam tip: For \(x^2=a\), the solutions are usually \(\pm\sqrt{a}\); for the positive value, take only \(\sqrt{a}\).

Evaluate the exponents first: \(7^2=49\) and \(5^2=25\). Therefore, \(7^2+5^2=49+25=74\). Option 64 may result from an incorrect evaluation of the squares. Exam tip: Find each square separately before adding.

First evaluate the powers: \(8^2=64\) and \(2^3=8\). Therefore, \(8^2-2^3=64-8=56\). The value 48 may result from incorrectly taking \(2^3\) as 16. Exam tip: In expressions with exponents, evaluate the powers before addition or subtraction.

Using the law of exponents, subtract the exponents when dividing powers with the same base: \(9^2 \div 9^2 = 9^{2-2} = 9^0 = 1\). Therefore, the correct answer is 1. Although 81 is the value of \(9^2\), here \(9^2\) is divided by itself. Exam tip: remember that \(a^0=1\) for every non-zero number \(a\).

Use the power-of-a-power rule: \((a^m)^n=a^{mn}\). Hence, \((2^2)^3=2^{2\times3}=2^6=64\). The value 32 may result from adding exponents incorrectly; in this rule, the exponents are multiplied. Exam tip: an exponent outside brackets applies to the entire power inside the brackets.

Here, \(10^3=1000\) and \(10^2=100\). Therefore, \(10^3-10^2=1000-100=900\). The option 800 can result from evaluating the powers incorrectly. Exam tip: evaluate each power separately before subtracting.

Given \(a^3=216\), taking the cube root of both sides gives \(a=\sqrt[3]{216}=6\), because \(6^3=6\times6\times6=216\). Option 5 is not correct because \(5^3=125\), not 216. Exam tip: To check a cube, multiply the number by itself three times.

First evaluate the powers: \(11^2=121\) and \(3^2=9\). Therefore, \(11^2+3^2=121+9=130\). Hence, option C is correct. 129 results from an addition error. Exam tip: evaluate each power first, then check the addition separately.

First evaluate the squares: \(12^2=144\) and \(4^2=16\). Therefore, \(12^2-4^2=144-16=128\), so 128 is correct. An answer such as 124 may result from an arithmetic error in subtraction. Exam tip: evaluate exponents before performing addition or subtraction.

When powers with the same base are divided, subtract the exponents: \(2^7 \div 2^5 = 2^{7-5} = 2^2 = 4\). Therefore, 4 is correct. Getting 8 results from not applying the exponent rule correctly. Exam tip: subtract exponents for division with the same base; add them for multiplication.

Here, 14^2 means 14 × 14, which equals 196. Therefore, 14^2 + 1 = 196 + 1 = 197. The option 196 is only the value of 14^2; the addition of 1 is still required. Exam tip: Evaluate the exponent first, then perform the operation outside it.

First evaluate the exponent: \(15^2=15\times15=225\). Then \(225-25=200\), so the correct answer is 200. Getting 210 indicates an error in subtraction. Exam tip: evaluate powers first, then perform subtraction.

Reason: Use the difference of squares identity \(a^2 - b^2 = (a-b)(a+b)\). With \(a=17\) and \(b=8\), \((17^2 - 8^2) = (17-8)(17+8) = 9\times25 = 225\). Brief notes on wrong options: option A (81) results from the mistake of computing \((17-8)^2\); option B (353) is the sum \(17^2+8^2\); option D (233) is a likely arithmetic subtraction error. Exam tip: Recognise and apply the identity \(a^2-b^2=(a-b)(a+b)\) to simplify such problems quickly and avoid careless subtraction mistakes.

18^2 = 324 and 19^2 = 361, so 324 + 361 = 685. Alternatively use the formula for consecutive squares: \(n^2+(n+1)^2=2n(n+1)+1\). With n = 18, \(2\times18\times19+1=684+1=685\). A close distractor like 675 typically arises from an arithmetic slip computing one of the squares. Exam tip: use the consecutive-square formula to avoid calculation errors under time pressure.

Since the square of 15 is 225, \(15^2=225\). Thus, \(b^2=225\) gives \(b=\pm15\), but the question asks for the positive value, so the answer is 15. Option 25 may seem tempting, but \(25^2=625\), not 225. Exam tip: when the positive value in \(x^2=a\) is asked, take \(\sqrt{a}\).

Apply the exponent first: \(20^2=400\). Adding 5 gives \(400+5=405\). Why other choices are wrong: 395 likely comes from mistakenly subtracting 5 (\(400-5\)); 415 would arise if 15 were added instead of 5; 420 could come from confusing the expression with \(20\times21\). Exam tip: evaluate powers before addition and double-check simple arithmetic to avoid common slip-ups.

First find the square of 21: \(21^2=441\). Then \(441-41=400\), so the correct answer is 400. Option 410 can result from an error in subtraction. Exam tip: In an expression with exponents, evaluate the exponent before performing addition or subtraction.

Here, 5^3 = 125 and 5^2 = 25. Therefore, 5^3 + 5^2 = 125 + 25 = 150. The option 125 is only the value of 5^3; 5^2 still has to be added. Exam tip: Evaluate each power separately before performing addition or subtraction.