Exponents

Using the laws of exponents, \((5^3)^2=5^{3\times2}=5^6\) and \(25^2=(5^2)^2=5^4\). Therefore, \(5^6\div5^4=5^{6-4}=5^2=25\). Hence, option A is correct. In such questions, first express all numbers as powers of the same base.

Evaluate the powers first: \(12^2=144\) and \(5^2=25\). Thus, \(12^2-5^2=144-25=119\), and \(119\div7=17\). Therefore, the correct answer is 17. Exam tip: follow the order of operations—powers first, then subtraction, and division last.

First evaluate the powers: 2^8 = 256 and 3^6 = 729. Adding them gives 256 + 729 = 985, so option B is correct. As an exam tip, calculate each power separately before performing the addition to avoid arithmetic errors.

When powers with the same base are multiplied, their exponents are added; hence \(a^m\times a^n=a^{m+n}\) is correct. Option B incorrectly multiplies the exponents. Exponents are multiplied when a power is raised to another power: \((a^m)^n=a^{mn}\), so option D is also incorrect. Exam tip: add exponents for multiplication of like bases, but multiply them for a power of a power.

Using the power-of-a-power rule, \((3^2)^3=3^{2\times3}=3^6\). For division with the same base, subtract the exponents: \(3^6\div3^4=3^{6-4}=3^2=9\). Therefore, the correct answer is 9. Exam tip: simplify the nested power first, then subtract exponents when dividing like bases.

Evaluate the powers first: 7^2 = 49 and 5^3 = 125. Therefore, 7^2 + 5^3 = 49 + 125 = 174, so option B is correct. In an exam, remember that exponents are evaluated before addition.

Evaluate the powers first: \(12^2=144\) and \(8^2=64\). Therefore, \(12^2-8^2=144-64=80\), so option C is correct. Exam tip: Check the subtraction order carefully; the expression can also be solved using the difference of squares, \((12-8)(12+8)=4\times20=80\).

Use the exponent rules \((a^m)^n=a^{mn}\) and \(a^m\div a^n=a^{m-n}\). Thus, \((5^2)^2\div 5^3=5^{2\times 2}\div 5^3=5^4\div 5^3=5^{4-3}=5\). Therefore, option B is correct. Option C, \(25=5^2\), can result from stopping after the power-of-a-power step or subtracting the exponents incorrectly. Exam tip: when dividing powers with the same base, subtract the exponents.

First evaluate the powers: 6^3 = 216 and 4^4 = 256. Adding them gives 216 + 256 = 472, so option C is correct. In an exam, calculate each power separately before performing the addition to avoid errors.

Use the difference of squares identity: \(a^2-b^2=(a-b)(a+b)\). Here \(15^2-11^2=(15-11)(15+11)=4\times26=104\). The closest distractor 94 is wrong — it would correspond to a subtraction or squaring error (for example misreading 121 as 131). Correct application of the identity gives 104. Exam tip: convert \(a^2-b^2\) to \((a-b)(a+b)\) to simplify mental calculation and avoid large subtractions.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\), we get \((2^5)^2=2^{10}\). For division with the same base, subtract the exponents: \(2^{10}\div2^8=2^{10-8}=2^2=4\). Therefore, the correct answer is 4. Exam tip: multiply exponents in a power raised to another power, then subtract exponents when dividing like bases.

Rewrite the denominator with the same base: \(9^2=(3^2)^2=3^4\). Therefore, \(3^6\div9^2=3^6\div3^4=3^{6-4}=3^2=9\). Hence, option C is correct. Exam tip: When powers with the same base are divided, subtract their exponents.

Since 16 = 2^4, we get 16^2 = (2^4)^2 = 2^8. Using the quotient rule for powers with the same base, 2^8 ÷ 2^6 = 2^(8−6) = 2^2 = 4. Therefore, the correct answer is 4. Exam tip: When dividing powers with the same base, subtract the exponents.

Square each number first and then add: \(14^2=196\) and \(6^2=36\). Therefore, \(14^2+6^2=196+36=232\), so option B is correct. Exam tip: evaluate the exponents before performing the addition; 242 is a close distractor but is not the sum of the correct squares.

According to the order of operations, evaluate the exponent first: \(3^2=9\). Then \(81\div 9=9\), so option C is correct. Option A (3) may result from confusing the base with the value of the expression, while option B ignores the division. Exam tip: In such expressions, calculate exponents before performing division.

Using the laws of exponents, \((10^3)^2=10^{3\times2}=10^6\). Therefore, \(10^6\div10^5=10^{6-5}=10\), so the correct answer is 10. The option 1000 results from incorrectly applying the division rule for powers with the same base. Exam tip: when dividing powers with the same base, subtract their exponents.

First evaluate the powers: \(2^8=256\) and \(5^3=125\). Therefore, \(2^8+5^3=256+125=381\), so option B is correct. In the exam, calculate the powers before performing addition or subtraction.

Apply the exponent laws: (6^2)^2=6^{2\times2}=6^4. Therefore, 6^4\div6^2=6^{4-2}=6^2=36. Hence, 36 is correct. The value 1296 is only 6^4, before dividing by 6^2. Exam tip: when powers with the same base are divided, subtract their exponents.

Evaluate the two powers first: 11² = 121 and 8² = 64. Therefore, 11² + 8² = 121 + 64 = 185, so option B is correct. In an exam, calculating each square separately before adding helps avoid arithmetic errors.

Since 32 = 2^5, we get 32^2 = (2^5)^2 = 2^{10}. Therefore, 2^{10}\div2^7 = 2^{10-7} = 2^3 = 8. Hence, option C is correct. Exam tip: When dividing powers with the same base, subtract their exponents.

Use the difference-of-squares identity: \(a^2-b^2=(a-b)(a+b)\). So \(19^2-15^2=(19-15)(19+15)=4\times34=136\). Thus the correct value is 136. Common mistakes: computing one square incorrectly (giving 126 on subtraction) or failing to add correctly; 146 is also incorrect because the correct sum is \(19+15=34\), not a larger value. Exam tip: convert differences of squares to a product to avoid large multiplications and reduce errors.

The zero-exponent rule is valid only when \(a\ne0\). For example, \(7^0=1\), but \(0^0\) is considered undefined in school mathematics. Therefore, base 0 is excluded. Exam tip: always check the non-zero-base condition.

First evaluate the powers: 3^5 = 243 and 4^4 = 256. Therefore, 3^5 + 4^4 = 243 + 256 = 499, so option B is correct. In an exam, calculate each power separately before performing the addition to avoid arithmetic errors.

Since \(243=3^5\), equal bases have equal exponents, so \(2x-1=5\). Thus, \(2x=6\) and \(x=3\). Therefore, option B is correct. Exam tip: Express the given number as a power of the same base before comparing exponents.

For division of powers with the same base, use \(\frac{a^m}{a^n}=a^{m-n}\). Thus, \(\frac{5^8}{5^3}=5^{8-3}=5^5\), so option C is correct. In option A, the exponent has been incorrectly taken as 6, while \(25^3=5^6\), making option D incorrect as well. In an exam, remember to subtract exponents only when the bases are the same.