Exponents

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.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((2^4)^3=2^{4\times3}=2^{12}\), so option A is correct. Remember that the exponents are multiplied, not added; hence \(2^7\) in option B is incorrect.

When powers with the same base are multiplied, their exponents are added: \(7^x \times 7^2=7^{x+2}\). Hence, \(x+2=9\), so \(x=7\), making option C correct. Exam tip: use \(a^m\times a^n=a^{m+n}\) for equal bases; do not multiply the exponents.

When powers with the same base are divided, their exponents are subtracted: \(\frac{2^{-3}}{2^{-5}}=2^{-3-(-5)}=2^2\). Therefore, option B is correct. Option C, \(2^{-2}\), results from subtracting the exponents in the wrong order. Exam tip: write \(-(-5)=+5\) explicitly when simplifying negative exponents.

Express both numbers with base 3: (9^x)=(3^2)^x=3^{2x} and 27=3^3. Thus, 3^{2x}=3^3, so 2x=3 and x=\(\frac{3}{2}\). Therefore, option B is correct. Exam tip: In exponential equations, rewrite both sides with the same base and then equate the exponents.

For powers with the same base, exponents are added during multiplication and subtracted during division: \(5^2 \times 5^3 \div 5^4 = 5^{2+3-4}=5^1=5\). Therefore, the correct answer is 5. The value 25 results from applying the exponent rules incorrectly. Exam tip: When the bases are the same, combine the exponents before evaluating the power.

Since \(64=4^3\), the equation becomes \(4^x=4^3\). With equal positive bases other than 1, their exponents are equal, so \(x=3\). In such questions, rewrite both sides with the same base before comparing exponents.

The negative-exponent law is \(a^{-n}=\frac{1}{a^n}\)。 Thus, \(\left(\frac{1}{3}\right)^{-4}=\left(\frac{3}{1}\right)^4=3^4=81\)。 Therefore, 81 is correct. The value \(\frac{1}{81}\) is incorrect because it would result from applying the positive fourth power without taking the reciprocal. Exam tip: for a negative exponent, first invert the base and then apply the positive exponent.

Using the law of fractional exponents, first take the fourth root of 16 and then cube the result: \(16^{3/4}=(16^{1/4})^3=2^3=8\). Therefore, option C is correct. Option A is only the fourth root \(16^{1/4}\), so it misses the cubing step. Exam tip: for \(a^{m/n}\), take the nth root first and then raise it to the mth power.

Since \(32=2^5\), the equation becomes \(2^{x+1}=2^5\). With equal bases, compare the exponents: \(x+1=5\), giving \(x=4\). Therefore, option B is correct. Remember not to take \(x=5\) directly because the exponent is \(x+1\).

Writing \(10=10^1\), the denominator becomes \(10^2\times10^1=10^{2+1}=10^3\). Therefore, \(\frac{10^6}{10^3}=10^{6-3}=10^3\), so option B is correct. Exam tip: for the same base, add exponents when multiplying and subtract them when dividing.

When powers with the same base are multiplied, their exponents are added: \(81^{1/2}\times81^{1/2}=81^{(1/2+1/2)}=81^1=81\). The value 9 represents only one factor, \(81^{1/2}\), not the complete product. Exam tip: use the rule \(a^m\times a^n=a^{m+n}\) for equal bases.

Since 25 can be written as a power of 5, \(25=5^2\). Therefore, \(25^x=(5^2)^x=5^{2x}\). In \(5^{2x}=5^6\), the bases are equal, so the exponents must be equal: \(2x=6\), giving \(x=3\). Hence, option B is correct. Exam tip: When exponential expressions have the same base, equate their exponents.