Exponents

Correct answer: A. \(2^{12}\)

Explanation: 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.

Correct answer: C. (7)

Explanation: 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.

Correct answer: B. \(2^2\)

Explanation: 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.

Correct answer: B. (\frac{3}{2})

Explanation: 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.

Correct answer: A. 5

Explanation: 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.

Correct answer: B. 3

Explanation: 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.

Correct answer: A. 81

Explanation: 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.

Correct answer: C. 8

Explanation: 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.

Correct answer: B. 4

Explanation: 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 \(x+1\).

Correct answer: B. 10^3

Explanation: 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.

Correct answer: C. 81

Explanation: 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.

Correct answer: B. (3)

Explanation: 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.

Correct answer: B. 4

Explanation: 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.

Correct answer: B. 7

Explanation: 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.

Correct answer: B. 16

Explanation: 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.

Correct answer: A. \(\frac{3}{2}\)

Explanation: 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.

Correct answer: B. 2

Explanation: 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.

Correct answer: A. \(\frac{2}{3}\)

Explanation: 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.

Correct answer: B. 4

Explanation: 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.

Correct answer: C. 49

Explanation: 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.

Correct answer: C. 6

Explanation: 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.

Correct answer: B. 1

Explanation: 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.

Correct answer: C. 25

Explanation: 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.

Correct answer: B. \(\frac{3}{5}\)

Explanation: 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.

Correct answer: A. 9

Explanation: 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.