Exponents

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.

Since 8=2^3, we get 8^x=(2^3)^x=2^{3x}. Thus 2^{3x}=2^{12}, so the exponents are equal: 3x=12, giving x=4. Exam tip: rewrite powers with a common base before comparing their exponents.

For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(\frac{7^4\times7^{-2}}{7}=7^{4+(-2)-1}=7^1=7\). Therefore, option B is correct. Exam tip: write the denominator 7 as \(7^1\) before applying the laws of exponents.

Using the rule for fractional exponents, \(64^{2/3}=(\sqrt[3]{64})^2\). Since \(\sqrt[3]{64}=4\), we get \(4^2=16\). Therefore, option B is correct. Option A is only the cube root and does not include the required squaring. Exam tip: For \(a^{m/n}\), it is often easiest to take the \(n\)th root first and then raise the result to the \(m\)th power.

Writing 125 as a power of 5 gives \(125=5^3\). Therefore, \(5^{2x}=5^3\), and since the bases are equal, their exponents must be equal: \(2x=3\). Hence, \(x=\frac{3}{2}\). Exam tip: When powers have the same base, equate their exponents directly.

Factor out \(3^4\) from the numerator: \(3^5-3^4=3^4(3-1)=2\cdot3^4\). Therefore, \(\frac{3^5-3^4}{3^4}=\frac{2\cdot3^4}{3^4}=2\), so option B is correct. Exam tip: rewrite \(3^5\) as \(3\cdot3^4\) before cancelling; cancelling terms across a subtraction would incorrectly give option 1.

Express both numbers as powers of 3: \(27=3^3\) and \(9=3^2\). Thus, \(27^x=9\) becomes \((3^3)^x=3^2\), so \(3^{3x}=3^2\). Since the bases are equal, their exponents must be equal: \(3x=2\), giving \(x=\frac{2}{3}\). Therefore, option A is correct. Option B, \(\frac{3}{2}\), results from reversing the comparison of the exponents. Exam tip: Rewrite both sides with the same base before comparing exponents.

Since \(4=2^2\), we have \(4^5=(2^2)^5=2^{10}\). Therefore, \(\frac{4^5}{2^8}=\frac{2^{10}}{2^8}=2^{10-8}=2^2=4\). Hence, option B is correct. Exam tip: when dividing powers with the same base, subtract the exponents; choosing 8 results from applying this rule incorrectly.

Here, (9^{1/2}=\sqrt{9}=3) and (16^{1/2}=\sqrt{16}=4). Therefore, ((3+4)^2=7^2=49). Hence, 49 is correct. Exam tip: convert an exponent of 1/2 into a square root first, simplify the terms, and then square their sum.

Since 16 = 2^4, the equation becomes 2^{x-2} = 2^4. Powers with the same base are equal only when their exponents are equal, so x - 2 = 4 and x = 6. Therefore, option C is correct. Exam tip: Rewrite the number on the right as a power with the same base before comparing exponents.

Using the laws of exponents for the common base 6, add the exponents in the numerator and subtract the exponent in the denominator: \(3+(-1)-2=0\). Therefore, \(6^0=1\), so option B is correct. Remember that the zero power of any non-zero number is 1.

Using the rule for fractional exponents, \(125^{2/3}=(\sqrt[3]{125})^2\). Since \(\sqrt[3]{125}=5\), the value is \(5^2=25\). Therefore, option C is correct. Exam tip: For \(a^{m/n}\), it is often easiest to take the \(n\)th root first and then raise the result to the \(m\)th power.

Express both numbers as powers of the same base: \(32=2^5\) and \(8=2^3\). Thus, \((2^5)^x=2^3\), giving \(2^{5x}=2^3\). Since the bases are equal, compare the exponents: \(5x=3\), so \(x=\frac{3}{5}\). Therefore, option B is correct. Exam tip: In exponential equations, rewrite both sides with the same base and then equate their exponents.

Since \(9=3^2\), we have \(9^{-1}=3^{-2}\). Using the division law for powers with the same base, \(a^m\div a^n=a^{m-n}\), we get \(3^{-2}\div3^{-4}=3^{-2-(-4)}=3^2=9\). Therefore, option A is correct. In exams, take special care that subtracting \(-4\) becomes addition of 4.