Exponents

When powers with the same non-zero base are divided, their exponents are subtracted: 2^8 ÷ 2^3 = 2^(8−3) = 2^5 = 32. Therefore, 32 is correct. The value 16 would result from incorrectly taking the exponent difference as 4. Exam tip: a^m ÷ a^n = a^(m−n), where a ≠ 0.

When powers with the same base are multiplied, their exponents are added: \(a^m \times a^n=a^{m+n}\). Therefore, \(3^5 \times 3^2=3^{5+2}=3^7=2187\). The distractor 729 equals \(3^6\), which results from adding one exponent incorrectly. Exam tip: for multiplication of powers with the same base, add the exponents and keep the base unchanged.

Use the power-of-a-power rule: \((a^m)^n = a^{mn}\). So \((5^2)^3 = 5^{2\times3} = 5^6\). Calculate \(5^6 = 15625\), hence option C is correct. Closest distractor 3125 equals \(5^5\) (one exponent short); 625 is \(5^4\) and 25 is \(5^2\), so they are incorrect. Exam tip: always simplify exponents using laws first, then evaluate the base power to avoid arithmetic mistakes.

Since \(14^3=14×14×14=2744\), it follows that \(x^3=2744\) gives \(x=\sqrt[3]{2744}=14\). The cubes of the other options are not equal to 2744. Exam tip: identify the nearest perfect cube first; here, \(14^3=2744\).

When powers with the same base are multiplied, their exponents are added: \\(2^5 \times 2^4 = 2^{5+4}=2^9=512\\). Therefore, the correct answer is 512. Remember that adding exponents is valid here because the bases are the same; 128, which is \\(2^7\\), is not correct.

When powers with the same base are divided, their exponents are subtracted: \(9^3 \div 9^1 = 9^{3-1}=9^2=81\). Therefore, option D is correct. Option A uses only the base, while option C is the value of \(9^3\) itself. Exam tip: remember the rule \(a^m \div a^n=a^{m-n}\).

Since 27 × 27 = 729, the positive value of a satisfying a² = 729 is 27. The negative value −27 also gives 729 when squared, but the question specifically asks for the positive value. Exam tip: the positive square root of 729 is √729 = 27.

An exponent of 3 means multiplying the number by itself three times. Thus, \(10^3=1000\) and \(5^3=125\). Therefore, \(10^3+5^3=1000+125=1125\), so option C is correct. In an exam, evaluate each power first and then add; do not treat \(10^3+5^3\) as \((10+5)^3\).

Use the law of exponents: for the same base, division subtracts exponents, i.e. \(a^m\div a^n=a^{m-n}\). Thus \(8^4\div 8^2=8^{4-2}=8^2=64\). About distractors: 512 = \(8^3\) (would result from an incorrect subtraction like 4−1), and 4096 = \(8^4\) (result if division is omitted). Exam tip: apply exponent rules first, then compute the remaining power to avoid arithmetic mistakes.

An exponent of 2 means multiplying the number by itself. Thus, 11^2 = 11 × 11 = 121 and 12^2 = 12 × 12 = 144. Therefore, 121 + 144 = 265, so option A is correct. In an exam, calculate each power first and then perform the addition to avoid errors.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(6^5 \div 6^3=6^{5-3}=6^2=36\). Option C is only the value of \(6^3\), so it is not correct. Exam tip: for division with the same base, subtract the exponents.

Given \(b^3=4096\), the value of \(b\) is the cube root of 4096. Since \(16^3=16\times16\times16=4096\), we get \(b=16\). Exam tip: remember common perfect cubes, including \(16^3=4096\), to identify the answer quickly.

Compute squares first: \(12^2=144\) and \(13^2=169\). Adding gives \(144+169=313\). The nearest distractor 312 likely comes from an off-by-one addition error; 323 and 303 reflect incorrect square values. Exam tip: find each square separately and check the unit digit (4+9=13 ⇒ unit digit 3 with a carry of 1) to verify quickly.

First evaluate the powers: 5^4 = 625 and 5^2 = 25. Therefore, 5^4 - 5^2 = 625 - 25 = 600, so option A is correct. Exam tip: In subtraction involving powers, evaluate each power before performing the subtraction.

Evaluate each square first: 13^2 = 13 × 13 = 169 and 14^2 = 14 × 14 = 196. Therefore, 13^2 + 14^2 = 169 + 196 = 365, so option B is correct. Exam tip: calculate the powers before performing the addition.

Evaluate the powers first: \(2^7=128\) and \(3^5=243\). Their sum is \(128+243=371\), so option C is correct. An exponent indicates how many times the base is multiplied by itself; calculating each power separately before addition is a reliable exam method.

Compute directly: \(14^2=196\) and \(15^2=225\), so \(196+225=421\). Alternate check: use identity \(a^2+b^2=(a+b)^2-2ab\); here \((14+15)^2-2\cdot14\cdot15=29^2-420=841-420=421\). The nearest wrong choice 411 can result from mistakenly taking \(15^2=215\) (giving 196+215=411). Exam tip: memorize common squares and use the identity to reduce calculation errors on paper.

Using the power-of-a-power rule, (a^m)^n = a^(mn), we get (2^4)^2 = 2^(4×2) = 2^8 = 256. Therefore, 256 is correct. The values 128 and 64 correspond to different powers, while 512 is 2^9. Exam tip: when a power is raised to another power, multiply the exponents.

First calculate the squares: 15^2 = 225 and 16^2 = 256. Adding them gives 225 + 256 = 481, so 481 is correct. Remember to evaluate each exponent before performing the addition.

Since (31)^2 = 31 × 31 = 961, the positive value of c is 31. Although c^2 = 961 gives c = 31 or c = -31, the question specifically asks for the positive value. Exam tip: when a positive value is requested from a square, choose the positive square root.

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n = a^{m-n}\). Thus, \(3^6 \div 3^3 = 3^{6-3} = 3^3 = 27\). Option B results from incorrectly using \(3^2\). In an exam, remember to subtract exponents when dividing powers with the same base.

First calculate both squares: \(17^2=289\) and \(18^2=324\). Therefore, \(17^2+18^2=289+324=613\), so option B is correct. Exam tip: Calculate each square separately before adding to avoid arithmetic errors.

Evaluate the two powers separately: 4^5 = 1024 and 2^8 = 256. Therefore, 4^5 + 2^8 = 1024 + 256 = 1280, so option C is correct. In an exam, calculate each exponent before performing the addition to avoid confusing the bases or exponents.

Given d^3 = 5832. Since 18 × 18 × 18 = 5832, the value of d is 18, so option A is correct. For an equation involving a cube, find the unknown by taking the cube root: d = ∛5832.

First calculate the two squares: 19^2 = 361 and 20^2 = 400. Therefore, 19^2 + 20^2 = 361 + 400 = 761, so option B is correct. In an exam, calculate each exponent separately before adding the terms to avoid confusing addition of squares with the square of a sum.