Exponents

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.