Operations on real numbers and the laws of exponents

Correct answer: A. (,4,)

Explanation: By exponent laws, ((2^3)^2=2^6) and (2^6 \times 2^{-4}=2^2=4). In exams, add exponents when the base is the same.

Correct answer: A. (,\dfrac{16}{3},)

Explanation: Here (5^0=1), (3^{-1}=\dfrac{1}{3}), and (2^{-2}=\dfrac{1}{4}), so the value is (\dfrac{16}{3}). In exams, first convert negative exponents into fractions.

Correct answer: A. (,a^2,)

Explanation: The numerator gives (a^m \times a^{2m}=a^{3m}), and then (\dfrac{a^{3m}}{a^{3m-2}}=a^2). In exams, subtract exponents during division.

Correct answer: A. (,\dfrac{x^4}{y^6},)

Explanation: The outside power (-2) multiplies both exponents, so (x^4y^{-6}=\dfrac{x^4}{y^6}). In exams, apply the outside power to every factor inside the bracket.

Correct answer: A. (,27,)

Explanation: Since (9^{\frac{3}{2}}=(\sqrt{9})^3=3^3=27). In exams, connect the exponent (\dfrac{1}{2}) with square root.

Correct answer: A. (,\dfrac{1}{8},)

Explanation: Here (16^{\frac{1}{4}}=2), so (16^{\frac{3}{4}}=8) and (16^{-\frac{3}{4}}=\dfrac{1}{8}). In exams, a negative exponent means reciprocal.

Correct answer: A. (,4\sqrt{2},)

Explanation: Because (\sqrt{50}=5\sqrt{2}), (\sqrt{8}=2\sqrt{2}), and (\sqrt{18}=3\sqrt{2}), the answer is (4\sqrt{2}). In exams, combine only like surd terms.

Correct answer: A. (,\sqrt{3}+\sqrt{2},)

Explanation: Multiplying by (\sqrt{3}+\sqrt{2}) makes the denominator (3-2=1). In exams, remember to multiply by the conjugate.

Correct answer: A. (,1,)

Explanation: \(\left(\dfrac{2}{3}\right)^{-2}=\left(\dfrac{3}{2}\right)^2=\dfrac{9}{4}\), so the product is (1). In exams, a fraction is inverted under a negative exponent.

Correct answer: A. (,27,)

Explanation: Here (27^{\frac{2}{3}}=9) and (81^{\frac{1}{4}}=3), so the product is (27). In exams, first take the root and then apply the power.

Correct answer: A. (,5,)

Explanation: From (9=3^2), (a=2), and from (8=2^3), (b=3), so (a+b=5). In exams, remembering small powers gives faster solutions.

Correct answer: A. (,a,)

Explanation: Inside, (a^{\frac{1}{2}}a^{\frac{3}{2}}=a^2), so (\dfrac{(a^2)^2}{a^3}=a). In exams, solve fractional exponents using the usual exponent rules.

Correct answer: A. (,x+y,)

Explanation: Because (x^2-y^2=(x-y)(x+y)), the simplified form is (x+y). In exams, identifying difference of squares is very useful.

Correct answer: A. (,4pq,)

Explanation: On expansion, ((p+q)^2=p^2+2pq+q^2) and ((p-q)^2=p^2-2pq+q^2), so the difference is (4pq). In exams, apply standard identities directly.

Correct answer: A. (,-6x^3y^3,)

Explanation: The product of coefficients (2) and (-3) is (-6), and powers of like variables are added. In exams, watch both the sign and the exponents carefully.

Correct answer: A. (,3a^2b^3,)

Explanation: The coefficient is (\dfrac{6}{2}=3), (a^{3-1}=a^2), and (b^{2-(-1)}=b^3). In exams, the sign changes when subtracting a negative exponent.

Correct answer: A. (,x^3,)

Explanation: The numerator is ((x^3)^2=x^6) and the denominator is (x^{-1}x^4=x^3), so the answer is (x^3). In exams, apply an exponent law at each step.

Correct answer: A. (,32,)

Explanation: Since (0.00032=3.2\times 10^{-4}), (\dfrac{3.2\times 10^{-4}}{10^{-5}}=3.2\times 10^1=32). In exams, converting decimals to scientific notation helps.

Correct answer: A. (,-3,)

Explanation: Since (125=5^3), (\dfrac{1}{125}=5^{-3}), so (n=-3). In exams, connect a reciprocal with a negative exponent.

Correct answer: A. (,6,)

Explanation: Taking (7^4) common in the numerator gives (\dfrac{7^4(7-1)}{7^4}=6). In exams, taking a common factor makes calculation shorter.

Correct answer: A. (,4,)

Explanation: The numerator is (2^{10}+2^{10}=2\times 2^{10}=2^{11}), so (\dfrac{2^{11}}{2^9}=2^2=4). In exams, first combine like terms and then apply exponent laws.

Correct answer: A. (,x^2-x-6,)

Explanation: Using the distributive law, (x(x-3)+2(x-3)=x^2-x-6). In exams, check the sign of the middle term carefully.

Correct answer: A. (,x^2+2x+4,)

Explanation: Because (x^3-8=(x-2)(x^2+2x+4)), the quotient is (x^2+2x+4). In exams, remember the identity for cubes.

Correct answer: A. 1

Explanation: Substituting \(x=2\), we get \(P(2)=2^3-4(2)+1=8-8+1=1\). Hence, option A is correct. Option B results from forgetting the final \(+1\). Exam tip: substitute the value carefully and simplify each term, especially the term containing the exponent.

Correct answer: A. (,18,)

Explanation: (\sqrt{12}\times \sqrt{27}=\sqrt{324}=18). In exams, use (\sqrt{a}\sqrt{b}=\sqrt{ab}) for non-negative numbers.