Exponents

Compute the powers separately: \(18^2=324\) and \(7^3=343\). Adding them gives \(324+343=667\), so 667 is correct. The close distractor 657 is wrong — it would result from mistakenly taking \(7^3\) as 333 (324+333=657). Exam tip: evaluate each power first, write intermediate results, then add to avoid arithmetic mistakes.

First evaluate the powers: \(12^3=1728\) and \(12^2=144\). Therefore, \(12^3-12^2=1728-144=1584\), so option A is correct. In an exam, calculate the two powers separately before performing the subtraction.

Evaluate the powers first: 19^2 = 19 × 19 = 361 and 8^3 = 8 × 8 × 8 = 512. Therefore, 19^2 + 8^3 = 361 + 512 = 873, so option B is correct. In an exam, calculate each power separately before performing the final addition to avoid arithmetic errors.

Given \(e^2=1849\), the positive value of \(e\) is the positive square root of 1849. Since \(43^2=43\times43=1849\), we get \(e=43\). The closest distractor, 42, is incorrect because \(42^2=1764\), not 1849. Exam tip: when a variable has a positive square value, check the positive integer whose square equals the given number.

Compute powers first: \(20^2=400\) and \(9^3=729\). Adding gives \(400+729=1129\), so 1129 is correct. The closest distractor C (1128) likely comes from misreading \(9^3\) as 728 or a one-unit arithmetic slip, so it is incorrect. Exam tip: evaluate each power separately, then add and quickly check the units digit for a sanity check.

For division of powers with the same base, use \(a^m \div a^n=a^{m-n}\). Thus, \(13^5 \div 13^4=13^{5-4}=13^1=13\). Therefore, option A is correct. Remember that exponents are subtracted when dividing like bases, not multiplied; hence 169, which is \(13^2\), is not the answer.

Evaluate the powers first: 21^2 = 21 × 21 = 441 and 10^3 = 10 × 10 × 10 = 1000. Therefore, 21^2 + 10^3 = 441 + 1000 = 1441, so option B is correct. In an exam, calculate each power separately before performing the addition.

Evaluate the powers first: \(2^8=256\) and \(3^7=2187\). Therefore, \(2^8+3^7=256+2187=2443\), so option C is correct. In an exam, calculate each power separately before performing the addition to avoid place-value errors.

Compute each power separately. \(22^2=22\times22=484\) and \(11^3=11\times11\times11=11\times121=1331\). Adding gives \(484+1331=1815\). Option B (1805) is wrong — it likely comes from a small arithmetic slip when computing the cube or the final addition. Exam tip: calculate powers one at a time and verify by checking last digits or re-adding to catch simple mistakes.

Given \(f^3=13824\). Since \(24^3=24\times24\times24=13824\), it follows that \(f=24\). The cubes of the other options are not equal to 13824. Exam tip: When the cube of a number is given, find its cube root and verify the answer by cubing it again.

Using the power-of-a-power rule, (a^m)^n = a^{mn}. Therefore, ((6^3)^2) = 6^{3×2} = 6^6 = 46656. Hence, 46656 is correct. Exam tip: when a power is raised to another power, multiply the exponents rather than adding them.

Evaluate the powers first: \(23^2=23\times23=529\) and \(12^3=12 imes12 imes12=1728\). Therefore, \(23^2+12^3=529+1728=2257\), so option C is correct. As an exam tip, calculate each exponent separately before performing the addition; adding the bases would be incorrect.

Compute the powers separately: \(24^2=576\) and \(13^3=2197\). Adding gives \(576+2197=2773\), so the correct value is 2773. The close distractor 2783 is incorrect — it reflects a typical addition or cube-miscalculation error (e.g. treating 2197 as 2207). Exam tip: calculate each power first and then add; memorise small squares and cubes to avoid simple mistakes.

For division of powers with the same base, subtract the exponents: \(a^m \div a^n=a^{m-n}\). Therefore, \(14^4 \div 14^2=14^{4-2}=14^2=196\). Option B, \(14^3\), would result only if the exponent difference were 3. Exam tip: subtract exponents for division with the same base and add them for multiplication.

Evaluate the powers first: 25^2 = 625 and 14^3 = 14 × 14 × 14 = 2744. Therefore, 25^2 + 14^3 = 625 + 2744 = 3369, so option B is correct. Exam tip: Calculate each exponent separately before performing the addition to avoid arithmetic errors.

Since \(45^2=45\times45=2025\), the positive value satisfying \(g^2=2025\) is \(g=45\). The equation has two real solutions, \(g=45\) and \(g=-45\), but the question specifically asks for the positive value. Exam tip: when the positive square root is requested, choose the principal square root.

Compute the powers first: \(26^2=676\) and \(15^3=3375\). Adding them gives \(676+3375=4051\), so 4051 is correct. The closest distractor 4041 likely results from a simple addition error (for example mis-adding by 10). Exam tip: evaluate each power separately, add using place value columns, and recheck the final addition to avoid small arithmetic mistakes.

For division of powers with the same base, use \(a^m \div a^n = a^{m-n}\). Thus, \(15^5 \div 15^4 = 15^{5-4} = 15^1 = 15\). Therefore, option A is correct. Remember that exponents are subtracted during division, not multiplied.

Evaluate the powers first: 27^2 = 27 × 27 = 729 and 16^3 = 16 × 16 × 16 = 4096. Therefore, 27^2 + 16^3 = 729 + 4096 = 4825, so option B is correct. Exam tip: calculate each power separately before performing the addition to avoid confusing exponentiation with addition.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(5^7 \div 5^4=5^{7-4}=5^3=125\), so option C is correct. Exam tip: For division with the same base, subtract the exponents rather than multiplying them.