Exponents

13^2=169 and 12^2=144, so 13^2-12^2=169-144=25. Alternatively, using the difference of squares identity a^2-b^2=(a-b)(a+b), we get (13-12)(13+12)=1×25=25. Therefore, option D is correct. Exam tip: For the difference between the squares of two close numbers, use the difference of squares identity for a quicker calculation.

When powers with the same base are multiplied, their exponents are added; hence \(a^m\times a^n=a^{m+n}\). Option C is incorrect because a power raised to a power gives \((a^m)^n=a^{mn}\). Exam tip: check whether the bases match first.

For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(3^4\times3^2\div3^5=3^{4+2-5}=3^1=3\). Therefore, option B is correct. Exam tip: apply these exponent laws directly only when the bases are the same.

When powers with the same non-zero base are divided, their exponents are subtracted: \\(a^m\div a^n=a^{m-n}\\). Therefore, \\(7^3\div7^2=7^{3-2}=7^1=7\\). Hence, option D is correct. Exam tip: for division with the same base, subtract the smaller exponent from the larger one; do not multiply the exponents.

Write 16 as a power of 4: 16 = 4^2, so 16^2 = (4^2)^2 = 4^4. Using the division law for powers with the same base, 4^4 ÷ 4^3 = 4^(4−3) = 4. Therefore, option A is correct. Exam tip: when dividing powers with the same non-zero base, subtract the exponent in the denominator from the exponent in the numerator.

Compute the powers first: \(2^6=64\) and \(3^4=81\). Adding them gives \(64+81=145\), so 145 is correct. Common wrong choices (e.g. 135 or 155) arise from mis-evaluating a power or a single-digit addition error. Exam tip: memorize small powers and recheck the final addition quickly to avoid slip errors.

Write 81 as a power of 3: \(81=3^4\). Using the quotient rule for powers with the same base, \(a^m\div a^n=a^{m-n}\), we get \(3^4\div3^3=3^{4-3}=3\). Therefore, option D is correct. Exam tip: when dividing powers with the same base, subtract the exponents; do not divide the base.

Using the laws of exponents, \((10^2)^2=10^{2\times2}=10^4\). For division with the same base, subtract the exponents: \(10^4\div10^3=10^{4-3}=10^1=10\). Therefore, the correct answer is 10. Option B is incorrect because it does not subtract the exponents correctly. Exam tip: multiply exponents in a power raised to another power, then subtract exponents when dividing like bases.

When powers with the same base are divided, their exponents are subtracted: \(a^m\div a^n=a^{m-n}\). Therefore, \(2^7\div2^2=2^{7-2}=2^5=32\). The distractor 64 may result from incorrectly adding the exponents. Exam tip: subtract exponents only when the bases are the same.

Using the power-of-a-power rule,
. Therefore,
and
. Hence, the value of
is 6. Exam tip: when powers with the same base are divided, subtract their exponents.

Since \(9=3^2\), we get \(9^2=(3^2)^2=3^4\). Therefore, \(3^5\div9^2=3^5\div3^4=3^{5-4}=3\). Hence, the correct answer is 3. Exam tip: when dividing powers with the same base, subtract the exponents; do not divide the bases.

Rewrite the terms with the same base: \(4=2^2\), so \(4^4=(2^2)^4=2^8\). Therefore, \(2^8\div2^6=2^{8-6}=2^2=4\). Hence, option B is correct. Exam tip: When dividing powers with the same non-zero base, subtract the exponents.

Evaluate the powers separately:
\(15^2=225\) and \(2^4=16\). Adding gives \(225+16=241\), so the correct value is 241.
Notes on distractors: B (233) reflects the mistake of treating \(2^4\) as 8 (which is \(2^3\)). A (225) corresponds to omitting the \(2^4\) term. D (256) is the value of a different power (for example \(4^4\) or \(16^2\)) and is not relevant here.
Exam tip: compute each power first, remember small powers of 2, then perform the addition to avoid slip errors.

For division of powers with the same base, use \(a^m\div a^n=a^{m-n}\). Thus, \(2^9\div2^7=2^{9-7}=2^2=4\), so option D is correct. Exam tip: subtract the exponents when dividing powers with the same non-zero base; do not multiply them.

Using the power-of-a-power rule, \((7^2)^3=7^{2\times3}=7^6\). When powers with the same base are divided, their exponents are subtracted: \(7^6\div7^5=7^{6-5}=7^1=7\). Therefore, the correct answer is 7. The nearby distractor 49 is the value of \(7^2\), but the remaining exponent here is 1. Exam tip: apply \((a^m)^n=a^{mn}\) first, followed by \(a^m\div a^n=a^{m-n}\).

Calculate the squares first: \(18^2=324\) and \(17^2=289\). Therefore, \(18^2-17^2=324-289=35\). It can also be solved using the identity \(a^2-b^2=(a-b)(a+b)\): \((18-17)(18+17)=1\times35=35\). In exams, use the difference-of-squares identity to avoid calculating large squares directly.

First evaluate the powers: 5^3 = 125 and 4^4 = 256. Adding them gives 125 + 256 = 381, so option C is correct. In an exam, calculate each power separately before adding the terms.

For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(2^5\times2^3\div2^4=2^{5+3-4}=2^4=16\), so option D is correct. Exam tip: when the base is the same, simplify the exponents before calculating the final value.

Write 8 as a power of 2: 8^2=(2^3)^2=2^6. Therefore, (2^6×2^3)÷2^7=2^(6+3−7)=2^2=4, so option C is correct. Option B (16) can result from incorrectly handling the exponents during multiplication or division. Exam tip: for powers with the same base, add exponents when multiplying and subtract them when dividing.

Evaluate the powers separately: \(11^2=121\) and \(6^3=216\). Therefore, \(11^2+6^3=121+216=337\), so option D is correct. In the exam, calculate the exponents before performing the addition.