Exponents

Correct answer: B. Irrational real number

Explanation: The decimal expansion is infinite and, according to the question, does not settle into any repeating block. A rational number must have a terminating decimal or an eventually repeating decimal, so this number is not rational. It is therefore an irrational real number. Option B is correct. An integer is rational, while a terminating decimal ends after finitely many places and cannot describe this expansion.

Correct answer: B. Irrational real number

Explanation: The key classification rule is that rational numbers have decimal expansions which terminate or eventually repeat. This expansion continues without end, and the question explicitly says that no fixed repeating block exists. It therefore represents an irrational real number, making option B correct. It cannot be an integer because every integer is rational, and it is not terminating because digits continue indefinitely.

Correct answer: A. 256

Explanation: When powers with the same base are multiplied, their exponents are added: \(2^5 \times 2^3 = 2^{5+3} = 2^8 = 256\). Therefore, 256 is correct. 128 equals \(2^7\), so it does not result from adding the exponents correctly. Exam tip: add exponents when multiplying like bases; subtract them when dividing.

Correct answer: B. 625

Explanation: When powers with the same base are divided, subtract the exponents: \(5^6 \div 5^2 = 5^{6-2}=5^4=625\). Therefore, 625 is correct. Getting 25 would result from subtracting the exponents incorrectly. Exam tip: for division with the same base, subtract the denominator’s exponent from the numerator’s exponent.

Correct answer: C. 729

Explanation: Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((3^2)^3=3^{2\times3}=3^6=729\). Hence, 729 is the correct answer. Note that 243 equals \(3^5\), so it is not correct. Exam tip: multiply the exponents when a power is raised to another power.

Correct answer: D. 1

Explanation: When powers with the same base are divided, subtract the exponents: \(10^4 \div 10^4 = 10^{4-4} = 10^0 = 1\). Therefore, the correct answer is 1. It is not 0, because any non-zero number divided by itself equals 1. Exam tip: use \(a^m \div a^n = a^{m-n}\), where \(a \ne 0\).

Correct answer: A. 343

Explanation: When powers with the same base are multiplied, their exponents are added: \(7^2 \times 7^1 = 7^{2+1} = 7^3 = 343\). Therefore, 343 is correct. The value 49 is only \(7^2\), so it is not the product. Exam tip: add exponents when multiplying powers with the same base.

Correct answer: B. 64

Explanation: When powers with the same base are divided, subtract the exponents: \(8^3 \div 8^1 = 8^{3-1} = 8^2 = 64\). Therefore, the correct answer is 64. The value 512 is \(8^3\), before dividing by \(8^1\). Exam tip: for division with the same base, subtract the exponents.

Correct answer: C. 64

Explanation: Using the exponent rule \((a^m)^n=a^{mn}\), \((2^3)^2=2^{3\times2}=2^6=64\). Therefore, 64 is correct. Getting 32 would incorrectly give the exponent as 5. Exam tip: when a power is raised to another power, multiply the exponents; do not add them.

Correct answer: D. 1

Explanation: For any non-zero number, the zero-exponent rule is \(a^0=1\), where \(a\ne0\). Therefore, \(9^0=1\). The value 9 is \(9^1\), not a number raised to the power 0. Exam tip: remember zero exponents using \(a^m/a^m=a^{m-m}=a^0=1\), rather than treating the exponent as multiplication by 0.

Correct answer: B. 1296

Explanation: When powers with the same base are multiplied, their exponents are added: \(6^2 \times 6^2 = 6^{2+2} = 6^4 = 1296\). Therefore, 1296 is correct. The value 36 is only \(6^2\), not the product of both terms. Exam tip: for multiplication with the same base, add the exponents.

Correct answer: A. 64

Explanation: When powers with the same base are multiplied, their exponents are added: \(2^4 \times 2^2 = 2^{4+2} = 2^6 = 64\). Therefore, 64 is correct. The value 32 equals \(2^5\), so it is not correct here. Exam tip: remember the rule \(a^m \times a^n = a^{m+n}\).

Correct answer: B. 9

Explanation: When powers with the same base are divided, subtract the exponents: \(3^5 \div 3^3 = 3^{5-3}=3^2=9\). Therefore, 9 is the correct answer. Getting 27 would result from applying the exponent rule incorrectly. Exam tip: remember \(a^m \div a^n=a^{m-n}\), where \(a\ne0\).

Correct answer: C. 625

Explanation: Using the rule \\((a^m)^n=a^{mn}\\), \\((5^2)^2=5^{2\times2}=5^4=625\\). Therefore, 625 is correct. The value 25 is only \\(5^2\\), before applying the outer exponent 2. Exam tip: when a power is raised to another power, multiply the exponents.

Correct answer: B. 1

Explanation: For any non-zero number, the zero-exponent rule is \(a^0=1\), where \(a\ne0\). Since \(100\) is non-zero, \(100^0=1\). It is not \(0\); an exponent of zero does not mean multiplying the number by zero. Exam tip: whenever you see \(a^0\), first check that the base is not zero.

Correct answer: C. 256

Explanation: Using the exponent rule
\((a^m)^n=a^{mn}\), we get
\((2^2)^4=2^{2\times4}=2^8=256\). Therefore, 256 is the correct answer. A value such as 64 can result from applying the rule incorrectly; when a power is raised to another power, the exponents are multiplied. Exam tip: in
\((a^m)^n\), always multiply
\(m\) and
\(n\).

Correct answer: D. 1

Explanation: When powers with the same base are divided, subtract their exponents: \(7^5 \div 7^5 = 7^{5-5} = 7^0 = 1\). Therefore, the correct answer is 1. The base 7 is not the result, and 0 is incorrect because any non-zero number raised to the power 0 equals 1. Exam tip: Use \(a^m \div a^m = a^0 = 1\), provided \(a \ne 0\).

Correct answer: A. 729

Explanation: When powers with the same base are multiplied, their exponents are added: \(3^2 \times 3^4 = 3^{2+4} = 3^6 = 729\). Therefore, 729 is correct. 243 equals \(3^5\), so it does not result from adding the exponents correctly. Exam tip: add exponents while multiplying like bases and subtract them while dividing.

Correct answer: B. 1000

Explanation: When powers with the same base are divided, subtract the exponents: \(10^6 \div 10^3 = 10^{6-3}=10^3=1000\). Therefore, 1000 is correct. The option 100 would result only if the exponent difference were 2. Exam tip: use \(a^m \div a^n=a^{m-n}\) only when the bases are the same and non-zero.

Correct answer: C. 148

Explanation: First evaluate the squares: \(12^2=144\) and \(2^2=4\). Therefore, \(12^2+2^2=144+4=148\). Hence, option C is correct. An answer such as 146 may result from an error while squaring or adding. Exam tip: calculate each square separately before adding the results.

Correct answer: D. 171

Explanation: The correct answer is 171. First find the squares: \(14^2=196\) and \(5^2=25\). Therefore, \(14^2-5^2=196-25=171\). A value such as 161 may result from an error in subtraction. Exam tip: calculate each square first and then subtract carefully.

Correct answer: A. \(\frac{a^m}{a^n}=a^{m-n}\)

Explanation: When powers with the same non-zero base are divided, subtract the exponent in the denominator from the exponent in the numerator: \(\frac{a^m}{a^n}=a^{m-n}\). Hence, option A is correct. Option D reverses the order of subtraction and is not generally correct. Exam tip: for division of like bases, use numerator exponent minus denominator exponent.

Correct answer: B. 100

Explanation: Evaluate the powers first: \(5^3=125\) and \(5^2=25\). Therefore, \(5^3-5^2=125-25=100\), so option B is correct. \(125\) is only the value of \(5^3\); it does not include subtracting \(5^2\). Exam tip: In expressions with exponents, evaluate each power before performing addition or subtraction.

Correct answer: C. 8

Explanation: When powers with the same base are divided, subtract the exponents: \(2^5 \div 2^2 = 2^{5-2} = 2^3 = 8\). Therefore, 8 is correct. Getting 16 may result from incorrectly adding the exponents. Exam tip: use \(a^m \div a^n = a^{m-n}\) for the same base.

Correct answer: D. 9

Explanation: When powers with the same base are divided, subtract the exponents: \(9^3 \div 9^2 = 9^{3-2} = 9^1 = 9\). Therefore, the correct answer is 9. The value 81 is \(9^2\) and may result from using the exponents incorrectly. Exam tip: use \(a^m \div a^n = a^{m-n}\) only when the bases are the same and non-zero.