Operations on real numbers and the laws of exponents

Correct answer: A. (,3,)

Explanation: (\dfrac{\sqrt{45}}{\sqrt{5}}=\sqrt{\dfrac{45}{5}}=\sqrt{9}=3). In exams, simplify division inside the root first.

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

Explanation: Because (\sqrt{a^2}=a) and (\sqrt{b^4}=b^2), the answer is (ab^2). In exams, note the condition that variables are positive.

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

Explanation: \(\dfrac{x^2}{y^{-1}}=x^2y\), so the whole square is \(x^4y^2\). In exams, simplify a negative exponent by moving its position.

Correct answer: A. (,-30,)

Explanation: The direct answer is option A, -30. Substitute a=2 and b=-3 carefully into a^2b-ab^2. First, a^2=2^2=4. Next, b^2=(-3)^2=9; the square is positive because two negative factors make a positive product. Therefore a^2b=4(-3)=-12 and ab^2=2(9)=18. The expression is a^2b-ab^2=-12-18=-30. Option A is correct. Option B, 6, does not follow from the substitution and usually comes from mishandling signs or powers. Option C, -6, is also incorrect because it does not equal the two calculated terms combined. Option D, 30, has the wrong sign; the second term is subtracted, so -12-18 remains negative. A reliable method is to calculate powers first, then multiplication, and finally the subtraction. The supplied answer is consistent.

Correct answer: A. (,24,)

Explanation: ((-2)^4=16) and ((-2)^3=-8), so (16-(-8)=24). In exams, odd and even powers have different signs.

Correct answer: A. (,-8,)

Explanation: A negative exponent inverts the fraction, so \(\left(-\dfrac{1}{2}\right)^{-3}=(-2)^3=-8\). In exams, keep the sign of a negative base according to the power.

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

Explanation: ((4x^{-2})^{-1}=4^{-1}x^2=\dfrac{x^2}{4}). In exams, apply the outside exponent to every factor of a product.

Correct answer: A. \(a^m\cdot a^n=a^{m+n}\)

Explanation: When powers with the same base are multiplied, their exponents are added, so \(a^m\cdot a^n=a^{m+n}\). In option B, exponents multiply: \((a^m)^n=a^{mn}\). Exam tip: first check whether the bases are the same before applying an exponent law.

Correct answer: A. (,14,)

Explanation: When the two squares are added, the surd terms cancel and (7+7=14). In exams, irrational terms often cancel in conjugate expressions.

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

Explanation: By the power of a power law, ((x^{\frac{1}{3}})^6=x^{\frac{6}{3}}=x^2). In exams, multiply the exponents.

Correct answer: A. (,\dfrac{a^3}{b^4},)

Explanation: In division, (a^{2-(-1)}=a^3) and (b^{-3-1}=b^{-4}), so the answer is (\dfrac{a^3}{b^4}). In exams, subtract exponents of like variables separately.

Correct answer: A. (,\dfrac{4}{9},)

Explanation: First, \(\left(\dfrac{81}{16}\right)^{\frac{1}{2}}=\dfrac{9}{4}\), then the negative exponent gives (\dfrac{4}{9}). In exams, check both the square root and the reciprocal.

Correct answer: A. (,10^8,)

Explanation: ((10^3)^2=10^6) and (\dfrac{10^6}{10^{-2}}=10^{6-(-2)}=10^8). In exams, be careful while subtracting a negative exponent.

Correct answer: A. (,\dfrac{26}{5},)

Explanation: Because (a^{-2}=\dfrac{1}{a^2}=\dfrac{1}{5}), (\dfrac{1}{5}+5=\dfrac{26}{5}). In exams, write (a^{-2}) as (\dfrac{1}{a^2}).

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

Explanation: (x^4-16=(x^2-4)(x^2+4)), so the simplified form is (x^2+4). In exams, treat (x^4) as ((x^2)^2) for factorisation.

Correct answer: A. (,2m^2+2n^2,)

Explanation: When both expansions are added, (2mn) and (-2mn) cancel, giving (2m^2+2n^2). In exams, notice opposite middle terms.

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

Explanation: Adding like terms gives (3x^2+x^2=4x^2), (-2x+5x=3x), and (1-4=-3). In exams, add like terms in columns.

Correct answer: A. (,3x^3-5x+12,)

Explanation: Changing the signs of the second bracket gives (5x^3-2x+7-2x^3-3x+5), so the answer is (3x^3-5x+12). In exams, change the sign of every term in the bracket during subtraction.

Correct answer: A. (,6x^2+5x-4,)

Explanation: The direct answer is A: 6x²+5x−4. Multiply every term in the first bracket by every term in the second bracket. First, 2x×3x=6x². Next, 2x×4=8x. Then, −1×3x=−3x. Finally, −1×4=−4. Combining like terms gives 6x²+8x−3x−4=6x²+5x−4. Option A is correct. Option B has the wrong sign for the x-term; the middle terms actually combine as 8x−3x=5x. Option C adds the cross terms as 8x+3x and therefore gets 11x, ignoring the negative sign. Option D has the wrong constant sign; (−1)(4)=−4, not +4. A useful check is that the leading coefficient is 2×3=6 and the constant is (−1)×4=−4. Memory cue: use FOIL carefully and preserve every sign.

Correct answer: A. (,\dfrac{x^6}{y^5},)

Explanation: (x^{5-(-1)}=x^6) and (y^{-2-3}=y^{-5}), so the form is (\dfrac{x^6}{y^5}). In exams, simplify the exponent of each variable separately.

Correct answer: A. \(,\dfrac{a^3}{b^4},\)

Explanation: The expression inside is \(a^{-3}b^4\), and the power (-1) gives its reciprocal \(\dfrac{a^3}{b^4}\). In exams, apply the outer negative power at the end.

Correct answer: A. (,\dfrac{1}{16},)

Explanation: Since (\sqrt[3]{64}=4), (4^{-2}=\dfrac{1}{16}). In exams, first evaluate the root and then apply the negative exponent.

Correct answer: A. (,\dfrac{6}{5},)

Explanation: (2^{-1}+3^{-1}=\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{5}{6}), so the whole value is (\dfrac{6}{5}). In exams, simplify the denominator first.

Correct answer: A. (,243,)

Explanation: Here (9^{-1}=3^{-2}) and (27^{-1}=3^{-3}), so the value is (3^{4-2-(-3)}=3^5=243). In exams, convert all terms to the same base.

Correct answer: A. (,\dfrac{b^5}{a^3},)

Explanation: The numerator is ((a^{-2}b^3)^2=a^{-4}b^6) and the denominator is ((ab^{-1})^{-1}=a^{-1}b), so the answer is (\dfrac{b^5}{a^3}). In exams, apply the outside power first.