Exponents

First find the square of 27: \(27^2=27\times27=729\). Then \(729-229=500\). Therefore, the correct answer is 500. A result such as 550 may come from an error in subtraction. Exam tip: evaluate the exponent first, then perform addition or subtraction.

First evaluate the powers: 2^7 = 128 and 2^5 = 32. Therefore, 2^7 - 2^5 = 128 - 32 = 96, so option A is correct. In an exam, calculating both powers before subtracting helps avoid errors; 64 would not be the result of this subtraction.

Since the positive value of (d) is required, take the positive square root: (d=\sqrt{529}=23). Checking gives (23^2=529). Although (-23) also satisfies the equation, it is not positive. In an exam, choose the positive root when the question asks for the positive value.

First evaluate the exponent: 29² = 29 × 29 = 841. Then 841 − 241 = 600, so the correct answer is 600. Exam tip: In such questions, evaluate the exponent or bracketed part first and then perform the subtraction.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(30^2 \div 30^1=30^{2-1}=30^1=30\), so option A is correct. Option B, 900, is only the value of \(30^2\), not the result after division. Exam tip: for division of powers with the same base, keep the base unchanged and subtract the exponents.

First evaluate the exponent: \(32^2=32\times32=1024\). Then subtract: \(1024-424=600\), so option C is correct. In such expressions, evaluate the exponent before performing subtraction.

First evaluate the exponent: \(33^2=33\times33=1089\). Then subtract: \(1089-389=700\). Therefore, option D is correct. Remember to calculate the power before carrying out the subtraction; do not use \(33-389\).

When powers with the same base are divided, their exponents are subtracted: 3^6 \div 3^4 = 3^{6-4} = 3^2 = 9. Therefore, 9 is correct. 27 is incorrect because it equals 3^3. Exam tip: remember that a^m \div a^n = a^{m-n}, where a ≠ 0.

We are given \(e^3=1331\). Since \(11^3=11\times11\times11=1331\), it follows that \(e=11\). In such questions, identifying the cube root of 1331 is the quickest method.

First evaluate the exponent: 35^2 = 35 × 35 = 1225. Then subtract 425: 1225 − 425 = 800, so option D is correct. Exam tip: evaluate exponents before carrying out addition or subtraction.

When powers with the same base are divided, their exponents are subtracted: \(4^5 \div 4^3=4^{5-3}=4^2=16\). Therefore, the correct answer is 16. Remember that exponents are subtracted, not added, when dividing like bases.

First evaluate the exponent: \(37^2=37\times37=1369\). Then, \(1369-769=600\), so option C is correct. In such questions, calculate the exponent before performing subtraction.

First evaluate the exponent: 38^2 = 38 × 38 = 1444. Then, 1444 − 644 = 800, so option D is correct. In such questions, remember to evaluate the exponent before performing subtraction.

When powers with the same base are divided, their exponents are subtracted: \(5^4 \div 5^2 = 5^{4-2} = 5^2 = 25\). Therefore, 25 is correct. Remember that exponents are subtracted in division and added in multiplication when the bases are the same.

First evaluate the exponent: \(41^2=41\times41=1681\). Then, \(1681+19=1700\). Therefore, option B, 1700, is correct. Exam tip: Evaluate the exponent before performing the addition and check the calculation in two steps.

It is given that the cube of f is 1728. Since 12 × 12 × 12 = 1728, f = 12. Therefore, option C is correct. Exam tip: For equations of the form f³ = n, find the cube root of n and check it by cubing the result.

First evaluate the exponent: \(42^2=42×42=1764\). Then subtract: \(1764-964=800\), so option D is correct. In the exam, follow the order of operations: calculate the exponent before subtraction.

First evaluate the exponent: \(43^2=43\times43=1849\). Then \(1849+151=2000\), so option B, 2000, is correct. In such questions, evaluate the exponent before carrying out addition or subtraction.

First evaluate the exponent: \(44^2=44\times44=1936\). Then \(1936-1136=800\), so option D is correct. In such questions, remember to calculate the power before performing the subtraction.

Since \(25^2=25\times25=625\), the positive value satisfying \(g^2=625\) is \(g=25\). Although \(-25\) also has square 625, the question specifically asks for the positive value. Exam tip: a positive number can have two square roots, but its positive square root is unique.