Exponents

When powers with the same base are multiplied, their exponents are added: \(2^6 \times 2^5=2^{6+5}=2^{11}\). Since \(2^{11}=2048\), option A is correct. Option B corresponds to \(2^{10}\), while option C corresponds to \(2^{12}\). Exam tip: remember the rule \(a^m \times a^n=a^{m+n}\) for multiplying powers with the same base.

When powers with the same non-zero base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Thus, \(3^8 \div 3^4=3^{8-4}=3^4=81\), so option B is correct. Exam tip: for division with the same base, subtract the exponents rather than multiplying them.

Using the power-of-a-power rule \\(a^m\\)^n = a^{mn}, we get \\(4^3\\)^2 = 4^{3×2} = 4^6 = 4096. Therefore, option C is correct. Exam tip: when a power is raised to another power, multiply the exponents rather than adding them.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(5^5 \div 5^3=5^{5-3}=5^2=25\), so option D is correct. Exam tip: For division with the same base, subtract the exponents rather than multiplying them.

Since \(21^3=21\times21\times21=441\times21=9261\), it follows that \(x=21\). Thus, the cube root of 9261 is 21. The cubes of the other options are not equal to 9261. Exam tip: identify the nearest perfect cube or verify the answer by cubing the option.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(6^6 \div 6^4=6^{6-4}=6^2=36\). Hence, option B is correct. Exam tip: for division with the same base, subtract the exponents rather than adding them.

First evaluate the powers: \(2^7=128\) and \(5^4=625\). Therefore, \(2^7+5^4=128+625=753\), so option C is correct. In an exam, calculate each power separately before performing the addition.

Since \(a^2=1521\), we have \(a=\pm\sqrt{1521}=\pm39\), because \(39^2=1521\). The question asks for the positive value, so \(a=39\). Remember that a positive perfect square has two square roots, but its positive square root is the positive one.

First evaluate the powers: 8^2 = 64 and 9^3 = 729. Then add them: 64 + 729 = 793, so option C is correct. Remember to evaluate the exponents before performing the addition.

For division of powers with the same base, use \(a^m \div a^n = a^{m-n}\). Therefore, \(9^5 \div 9^4 = 9^{5-4} = 9^1 = 9\). Hence, option D is correct. Exam tip: subtract the exponents when the bases are the same; do not subtract or divide the bases.

First evaluate the powers: \(3^6=729\) and \(3^3=27\). Therefore, \(3^6-3^3=729-27=702\), so option A is correct. Option B is only the value of \(3^6\) and does not subtract \(3^3\). Exam tip: In an expression involving powers, evaluate the powers first and then perform the indicated addition or subtraction.

Evaluate the powers first: \(10^2=100\) and \(11^3=11\times11\times11=1331\). Therefore, \(10^2+11^3=100+1331=1431\), so option B is correct. Exam tip: calculate exponents before performing addition.

Since b^3=10648, the value of b is the cube root of 10648. Checking the options, 22^3=22×22×22=484×22=10648; therefore, b=22 and option C is correct. In the exam, checking the cubes of the given options is a quick and reliable method.

Compute the powers separately: \(12^2=144\) and \(13^3=13\times13\times13=2197\). Adding gives \(144+2197=2341\). Option B (2331) is a common trap — it results if \(13^3\) is mistakenly taken as \(2187\) (confusing with \(3^7\)), producing \(144+2187=2331\). The other wrong choices arise from simple multiplication or addition errors. Exam tip: always calculate powers first and check the last digit to catch simple slip errors before finalizing the answer.

Using the power-of-a-power rule \\(a^m\\)^n=a^{mn}, we get \\((2^5)^2=2^{5\times2}=2^{10}=1024\\). Therefore, option A is correct. Remember that the exponents are multiplied, not added, in this rule.

Evaluate the powers first: 13^2 = 13 × 13 = 169 and 14^3 = 14 × 14 × 14 = 2744. Therefore, 169 + 2744 = 2913, so option B is correct. In an exam, calculate each power separately before performing the addition to avoid arithmetic errors.

Evaluate the powers first: 4^4 = 4 × 4 × 4 × 4 = 256 and 6^3 = 6 × 6 × 6 = 216. Therefore, 4^4 + 6^3 = 256 + 216 = 472, so option C is correct. In an exam, calculate each power separately before performing the addition.

First evaluate the powers: \(15^2=225\) and \(16^3=4096\). Adding them gives \(225+4096=4321\), so option D is correct. Exam tip: Calculate each exponent separately before performing the addition to avoid arithmetic errors.

Since (41)^2 = 41 × 41 = 1681, the possible values of c are 41 and −41. The question asks for the positive value, so the correct answer is 41. Exam tip: When the square of a number is given, take its square root and choose the positive sign when a positive value is required.

Evaluate the powers first: \(17^2=17×17=289\) and \(5^4=5×5×5×5=625\). Therefore, \(17^2+5^4=289+625=914\), so option C is correct. Option B results from an incorrect addition. Exam tip: calculate each power separately and add the results only at the end to avoid computation errors.

Compute the powers first: \(8^3=8\times8\times8=512\) and \(12^2=12\times12=144\). Adding them gives \(512+144=656\). Thus the correct value is 656. A common close distractor (646) results from arithmetic slip — for example miscomputing \(12^2\) or \(8^3\). Exam tip: evaluate each power separately, then add; use simple checks (like units digit or rough estimation) to catch calculation errors.

For division of powers with the same base, use the law \(a^m \div a^n = a^{m-n}\). Thus, \(11^4 \div 11^2 = 11^{4-2} = 11^2 = 121\). Option B, 1331, is the value of \(11^3\), so it is not correct. Exam tip: When dividing powers with the same base, subtract their exponents.

Given \(d^3=12167\), we calculate \(23^3=23\times23\times23=529\times23=12167\). Therefore, \(d=23\). The cubes of the other options are not equal to 12167. Exam tip: when the cube of a number is given, find its cube root and verify the answer by multiplying it three times.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((3^2)^4=3^{2\times4}=3^8=6561\), so option C is correct. Option A equals \(3^6\), option B equals \(3^7\), and option D equals \(3^4\). Exam tip: when a power is raised to another power, multiply the exponents rather than adding them.

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.