Exponents

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.

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.

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.

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.

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.

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 \(a^m \times a^n = a^{m+n}\).

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 \(a\ne0\).

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.

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.

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
\(n\).

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 \(a \ne 0\).

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.

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.

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.

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.

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.

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.

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.

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.

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.