Exponents

Correct answer: A. 256

Explanation: Using the exponent rule
((a^m)^n=a^{mn}), we get
((4^2)^2=4^{2\times2}=4^4=256). Therefore, 256 is correct. The value 16 is only
(4^2); the outer square must also be applied. Exam tip: when a power is raised to another power, multiply the exponents.

Correct answer: B. 10000

Explanation: Using the power-of-a-power rule, \((a^m)^n=a^{mn}\), we get \((10^2)^2=10^{2\times2}=10^4=10000\). Therefore, 10000 is correct. The value 100 is only \(10^2\); here, the entire \(10^2\) is squared again. Exam tip: when a power is raised to another power, multiply the exponents rather than adding them.

Correct answer: B. 729

Explanation: \(3^6\) means multiplying 3 by itself 6 times: \(3\times3\times3\times3\times3\times3=729\). Therefore, 729 is the correct answer. 243 is the value of \(3^5\), so it is a close but incorrect option. Exam tip: the exponent tells how many times the base is multiplied by itself.

Correct answer: A. 1

Explanation: For any positive integer exponent, multiplying 1 by itself any number of times still gives 1. Therefore, \(1^9=1\). The number 9 is the exponent, not the value, while 81 is the value of \(9^2\). Exam tip: every power of 1 is always 1.

Correct answer: B. 2

Explanation: Any non-zero number raised to the power 0 equals 1. Hence, 2^0=1 and 5^0=1, so 2^0+5^0=1+1=2. Option 1 is the value of only one term, not their sum. Exam tip: remember that a^0=1 for every non-zero base a.

Correct answer: C. 3

Explanation: Any non-zero number raised to the power 0 equals 1. Thus, \(3^0=1\), \(4^0=1\), and \(5^0=1\). Therefore, \(3^0+4^0+5^0=1+1+1=3\). Option 1 is the value of each individual term, not of the complete sum. Exam tip: remember that \(a^0=1\) for \(a\ne0\).

Correct answer: A. 43

Explanation: Evaluate the powers first: \(2^4=2\times2\times2\times2=16\) and \(3^3=3\times3\times3=27\). Hence, \(2^4+3^3=16+27=43\). Option 33 results from evaluating \(2^4\) incorrectly. Exam tip: In an expression with exponents, find each power before performing addition or subtraction.

Correct answer: B. 157

Explanation: First evaluate the powers: \(5^3=5\times5\times5=125\) and \(2^5=2\times2\times2\times2\times2=32\). Therefore, \(5^3+2^5=125+32=157\). Option 147 results from an incorrect evaluation of a power or the addition. Exam tip: In expressions with exponents, evaluate each power before performing addition or subtraction.

Correct answer: C. 136

Explanation: First evaluate each square: \(10^2=100\) and \(6^2=36\). Hence, \(10^2+6^2=100+36=136\), so option C is correct. A value such as 146 may result from incorrectly evaluating \(6^2\). Exam tip: simplify powers first, then perform addition or subtraction.

Correct answer: D. 144

Explanation: Use the identity \(a^2-b^2=(a-b)(a+b)\). Thus, \(15^2-9^2=(15-9)(15+9)=6\times24=144\). Therefore, 144 is the correct answer. A value such as 154 can result from an error in subtraction or multiplication. Exam tip: For expressions involving a difference of squares, use the identity instead of calculating both squares separately.

Correct answer: A. 73

Explanation: Evaluate the powers first: \(4^3=4\times4\times4=64\) and \(3^2=3\times3=9\). Therefore, \(4^3+3^2=64+9=73\). Option 91 would result from incorrectly treating \(3^2\) as \(3^3\). Exam tip: calculate each exponent separately before adding or subtracting.

Correct answer: B. 113

Explanation: Here, 7^2 = 49 and 8^2 = 64. Therefore, 49 + 64 = 113, so the correct answer is 113. Getting 103 would result from an error in evaluating the squares or adding them. Exam tip: find each square separately before adding the results.

Correct answer: A. 97

Explanation: Evaluate the powers first: \(9^2=81\) and \(4^2=16\). Therefore, \(9^2+4^2=81+16=97\). The value \(169\) is for \((9+4)^2\), which is not the given expression. Exam tip: evaluate each exponent term separately before performing addition or subtraction.

Correct answer: B. 55

Explanation: Evaluate the powers first: \(8^2=64\) and \(3^2=9\). Hence, \(8^2-3^2=64-9=55\). The value 65 results from an error in subtraction. Exam tip: In an expression with exponents, evaluate the powers before addition or subtraction.

Correct answer: A. 16

Explanation: When powers with the same base are divided, subtract the exponents: \(2^8 \div 2^4 = 2^{8-4} = 2^4 = 16\). Therefore, 16 is correct. Getting 8 would result from subtracting the exponents incorrectly or using \(2^3\). Exam tip: for division with the same base, use \(a^m \div a^n=a^{m-n}\).

Correct answer: B. 9

Explanation: When powers with the same base are divided, subtract the exponents: \(3^7 \div 3^5 = 3^{7-5}=3^2=9\). Therefore, 9 is correct. 27 equals \(3^3\), which does not use the correct difference of exponents. Exam tip: use \(a^m \div a^n=a^{m-n}\) only when the bases are the same and non-zero.

Correct answer: C. 1296

Explanation: Using the law of exponents \((a^m)^n = a^{mn}\), we have \((6^2)^2 = 6^{2\times2} = 6^4\). Compute stepwise: \(6^2=36\) and \(36^2=1296\), so the value is 1296. Distractors: 36 equals \(6^2\), 216 equals \(6^3\), and 256 is unrelated here. Exam tip: apply the power-of-a-power rule and simplify by computing smaller powers first if helpful.

Correct answer: B. 1

Explanation: Since 50 is non-zero, the zero-exponent rule applies: for any non-zero number \(a\), \(a^0=1\). Hence, \(50^0=1\). It is not 0; zero is the exponent, not the value of the expression. Exam tip: Before applying this rule, check that the base is not 0.

Correct answer: A. 298

Explanation: Here, 17^2 = 289 and 3^2 = 9. Therefore, 17^2 + 3^2 = 289 + 9 = 298. The value 289 is only 17^2; 3^2 must still be added. Exam tip: Evaluate each exponent term separately before performing addition or subtraction.

Correct answer: D. 275

Explanation: Use the difference of squares identity: \(18^2-7^2=(18-7)(18+7)=11\times25=275\). Therefore, 275 is the correct answer. Getting 265 usually results from an error in subtraction. Exam tip: use \(a^2-b^2=(a-b)(a+b)\) to calculate such expressions quickly.

Correct answer: A. 16

Explanation: When powers with the same base are divided, subtract the exponents: \(4^6 \div 4^4 = 4^{6-4} = 4^2 = 16\). Therefore, 16 is correct. The option 4 would result only if the difference of the exponents were 1. Exam tip: use \(a^m \div a^n = a^{m-n}\) only when the bases are the same and non-zero.

Correct answer: A. 2401

Explanation: Use the power-of-a-power rule: \((a^{m})^{n}=a^{m\times n}\). So \((7^2)^2=7^{2\times2}=7^4=2401\). Option B (343) equals \(7^3\), option C (49) equals \(7^2\), and option D (2400) is numerically close but incorrect. Exam tip: For (a^m)^n multiply the exponents m and n before evaluating the base.

Correct answer: A. 3125

Explanation: When powers with the same base are multiplied, their exponents are added: \(5^2 \times 5^3 = 5^{2+3} = 5^5 = 3125\). Therefore, 3125 is correct. The close distractor 625 equals \(5^4\), so it does not result from adding the exponents correctly. Exam tip: use \(a^m \times a^n = a^{m+n}\) only when the bases are the same.

Correct answer: B. 36

Explanation: When powers with the same base are divided, subtract the exponents: \(6^4 \div 6^2 = 6^{4-2}=6^2=36\). Therefore, 36 is correct. The value 6 is only \(6^1\), while 216 equals \(6^3\). Exam tip: remember \(a^m \div a^n=a^{m-n}\), where \(a\ne0\).

Correct answer: D. 64

Explanation: When powers with the same base are divided, subtract the exponents: \(8^5 \div 8^3=8^{5-3}=8^2=64\). Therefore, 64 is correct. 512 is the value of \(8^3\), not the result after subtracting the exponents. Exam tip: use \(a^m \div a^n=a^{m-n}\) only when the bases are the same and non-zero.