Operations on real numbers and the laws of exponents

Correct answer: A. (,\dfrac{27}{8},)

Explanation: \(\left(\dfrac{9}{4}\right)^{\frac{1}{2}}=\dfrac{3}{2}\), so \(\left(\dfrac{9}{4}\right)^{\frac{3}{2}}=\left(\dfrac{3}{2}\right)^3=\dfrac{27}{8}\). In exams, take the square root first.

Correct answer: A. (,\dfrac{8}{3},)

Explanation: Inside, (2^{-3}+2^{-2}=\dfrac{1}{8}+\dfrac{1}{4}=\dfrac{3}{8}), so the power (-1) gives (\dfrac{8}{3}). In exams, simplify the bracket first.

Correct answer: A. (,x^2-9,)

Explanation: (x^4-81=(x^2-9)(x^2+9)), so the simplified form is (x^2-9). In exams, treat (x^4) as ((x^2)^2) while factoring.

Correct answer: A. (,3x^2+4x-3,)

Explanation: Adding like terms gives (4x^2-x^2=3x^2), (-3x+7x=4x), and (6-9=-3). In exams, add like terms separately.

Correct answer: A. (,-3y^3-5y+13,)

Explanation: Changing all signs in the second bracket gives (2y^3-y+5-5y^3-4y+8). In exams, change the sign of every term during subtraction.

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

Explanation: On expansion, ((x+1)^3=x^3+3x^2+3x+1) and ((x-1)^3=x^3-3x^2+3x-1), so the difference is (6x^2+2). In exams, expand cubes carefully.

Correct answer: A. 28

Explanation: Using the laws of exponents, \(a^{2m+n}=a^{2m}\cdot a^n=(a^m)^2\cdot a^n\). Substituting the given values gives \(2^2\times 7=4\times 7=28\). Therefore, the correct answer is 28. In an exam, split a sum in the exponent into a product of powers; multiplying \(a^m\) and \(a^n\) directly would give 14, which corresponds to \(a^{m+n}\).

Correct answer: A. (,\dfrac{4x^2}{y},)

Explanation: ((64)^{\frac{1}{3}}=4), ((x^6)^{\frac{1}{3}}=x^2), and ((y^{-3})^{\frac{1}{3}}=y^{-1}), so the answer is (\dfrac{4x^2}{y}). In exams, apply the exponent to each factor.

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

Explanation: Because (\sqrt{a^4}=a^2) and (\sqrt{b^2}=b), the simplified form is (a^2b). In exams, note the positive condition.

Correct answer: A. (,3x^2+y^2,)

Explanation: The numerator difference is (6x^2y+2y^3=2y(3x^2+y^2)), so division gives (3x^2+y^2). In exams, take out the common factor.

Correct answer: A. (,243,)

Explanation: (9^2=3^4) and (27^{-1}=3^{-3}), so the value is (3^{-2+4-(-3)}=3^5=243). In exams, be careful while subtracting a negative exponent.

Correct answer: A. (,3,)

Explanation: The direct answer is option A: 3. Since a≠0, division of powers with the same nonzero base is allowed: \(\frac{a^m}{a^n}=a^{m-n}\). Therefore \(\frac{a^5}{a^k}=a^{5-k}\). The question says this equals a^2, so a^{5−k}=a^2. Because a≠1, equal powers here require equal exponents: 5−k=2. Subtract 2 from 5, or rearrange, to obtain k=3. Option A, 3, is correct. Option B, 2, would make the left side \(a^{5-2}=a^3\), not a^2. Option C, 5, would make the quotient \(a^0=1\), not a^2. Option D, 7, would give \(a^{-2}=1/a^2\), not a^2 in general. The conditions a≠0 and a≠1 prevent division by zero and ambiguity from the base 1. The memory rule is: when dividing equal bases, subtract the denominator exponent from the numerator exponent.

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

Explanation: The numerator is ((2x)^3(3x^{-2})=8x^3\cdot 3x^{-2}=24x), and (\dfrac{24x}{12x^{-1}}=2x^2). In exams, simplify both coefficient and variable parts.

Correct answer: A. (,8\sqrt{2},)

Explanation: (\sqrt{98}=7\sqrt{2}), (\sqrt{72}=6\sqrt{2}), and (\sqrt{50}=5\sqrt{2}), so the answer is (8\sqrt{2}). In exams, first write all surds in simplest form.

Correct answer: A. (,6+3\sqrt{3},)

Explanation: Multiplying by (2+\sqrt{3}) makes the denominator (4-3=1). In exams, multiply both numerator and denominator by the conjugate.

Correct answer: A. (,20,)

Explanation: (4^{-1}-5^{-1}=\dfrac{1}{4}-\dfrac{1}{5}=\dfrac{1}{20}), so the whole value is (20). In exams, first convert negative powers into fractions.

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

Explanation: Because (x^2+3x+2=(x+1)(x+2)), the simplified form is (x+2). In exams, factorise trinomials carefully.

Correct answer: A. (,5,)

Explanation: (125^{\frac{2}{3}}=25) and (25^{\frac{1}{2}}=5), so the value is (5). In exams, separate fractional exponents into root and power.

Correct answer: A. (,63,)

Explanation: Squaring both sides gives (n=(3\sqrt{7})^2=9\times 7=63). In exams, square both sides in a square root equation.

Correct answer: A. (,a+b,)

Explanation: The numerator is (a^{-1}+b^{-1}=\dfrac{a+b}{ab}) and the denominator is ((ab)^{-1}=\dfrac{1}{ab}), so the answer is (a+b). In exams, make a common denominator.

Correct answer: A. (,12,)

Explanation: ((2^5)^{\frac{2}{5}}=2^2=4) and ((3^3)^{\frac{1}{3}}=3), so the product is (12). In exams, apply the power of a power law.

Correct answer: A. \(\frac{x^{5}}{4}\)

Explanation: Here \(\left(2x^{-3}\right)^{-2}=2^{-2}x^{6}=\frac{x^{6}}{4}\), so multiplying by \(x^{-1}\) gives \(\frac{x^{5}}{4}\). In exams, first convert negative exponents carefully.

Correct answer: A. (a)

Explanation: The numerator exponent is ((m+2)+(3-m)=5), and (\frac{a^{5}}{a^{4}}=a). In exams, add and subtract exponents only for the same base.

Correct answer: B. (3)

Explanation: We have \(\left(3^{x}\right)^{2}=3^{2x}\) and \(729=3^{6}\), so \(2x=6\) and \(x=3\). In exams, rewrite both sides with the same base.

Correct answer: B. (5)

Explanation: Since (25^{-2}=5^{-4}) and (125=5^{3}), (\frac{5^{8}\cdot5^{-4}}{5^{3}}=5^{1}=5). In exams, convert (25) and (125) into powers of (5).