Operations on real numbers and the laws of exponents

Correct answer: A. \((7x-8)(7x+8)\)

Explanation: \(49x^2-64=(7x)^2-8^2\). Using the difference of squares identity \(a^2-b^2=(a-b)(a+b)\), we get \((7x-8)(7x+8)\). Option B does not expand to the given expression, while C and D are squares rather than the correct difference-of-squares factorization. Exam tip: first rewrite both terms as perfect squares.

Correct answer: A. \((x+7)^2\)

Explanation: Use the perfect-square identity \((a+b)^2=a^2+2ab+b^2\). Taking \(a=x\) and \(b=7\), we get \((x+7)^2=x^2+2\cdot x\cdot7+7^2=x^2+14x+49\), so option A is correct. Option B would produce the middle term \(-14x\), not \(+14x\). Exam tip: take the square root of the constant term and use the sign of the middle term to choose the appropriate binomial.

Correct answer: A. \((x-8)^2\)

Explanation: The expression can be written as \(x^2-2\cdot x\cdot 8+8^2\). Using \(a^2-2ab+b^2=(a-b)^2\), it equals \((x-8)^2\). Option B would produce a middle term of \(+16x\), so it is incorrect. Exam tip: identify the square roots of the first and last terms, then check whether the middle term is \(-2ab\) or \(+2ab\).

Correct answer: A. 45

Explanation: Use the identity \(m^2+n^2=(m+n)^2-2mn\). Thus, \(m^2+n^2=9^2-2(18)=81-36=45\), so the correct answer is 45. The option 63 may result from an incorrect calculation while subtracting \(2mn\). Exam tip: remember the identity \(a^2+b^2=(a+b)^2-2ab\) for such questions.

Correct answer: C. 37

Explanation: Use the identity \(a^2+b^2=(a-b)^2+2ab\). Thus, \(a^2+b^2=5^2+2(6)=25+12=37\). Option B results from incorrectly adding only \(ab\) instead of \(2ab\). In exams, remember that \((a-b)^2=a^2-2ab+b^2\), which leads to this identity.

Correct answer: A. 9801

Explanation: Since \(99=100-1\), \(99^2=(100-1)^2=100^2-2\times100\times1+1^2=10000-200+1=9801\). Option B does not correctly account for subtracting 200. In exams, express a number near 100 as \((a-b)\) to calculate its square quickly.

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

Explanation: When powers with the same base are multiplied, their exponents are added; hence \(a^m\times a^n=a^{m+n}\). Multiplying exponents to get \(mn\) is incorrect here. Exam tip: check that the bases match first.

Correct answer: A. 2491

Explanation: \(53\cdot47=(50+3)(50-3)\). Using the difference-of-squares identity \((a+b)(a-b)=a^2-b^2\), we get \(50^2-3^2=2500-9=2491\). Therefore, option A is correct. Option B, 2509, may result from adding 9 instead of subtracting it. Exam tip: When two numbers are equally above and below a common number, use the difference-of-squares identity.

Correct answer: B. \(13\)

Explanation: Using the negative-exponent rule \(a^{-n}=\frac{1}{a^n}\), we get \(\left(\frac{1}{3}\right)^{-2}=3^2=9\) and \(\left(\frac{1}{2}\right)^{-2}=2^2=4\). Therefore, the sum is \(9+4=13\). The value \(\frac{13}{36}\) results from mishandling the negative powers of the fractions. Exam tip: for a negative exponent, take the reciprocal of the base before applying the exponent.

Correct answer: B. 4

Explanation: For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(\dfrac{2^3\times2^4}{2^5}=2^{3+4-5}=2^2=4\). Therefore, the correct answer is 4. The value 8 may result from forgetting to subtract the exponent in the denominator. Exam tip: When the bases are the same, simplify the exponents before calculating the numerical value.

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

Explanation: Both terms, \(2x^3\) and \(6x^2\), contain 2 and \(x^2\), so the common factor is \(2x^2\). Dividing each term by it gives \(2x^3+6x^2=2x^2(x+3)\). Option B does not reproduce the second term correctly. Exam tip: take the HCF of the numerical coefficients and the lowest power of the common variable.

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

Explanation: Divide the numerical coefficients to get \(15\div5=3\). For like bases, subtract the exponents: \(a^{4-2}=a^2\) and \(b^{2-1}=b\). Therefore, the simplified form is \(3a^2b\). Option B is incorrect because it does not reduce the exponent of \(b\). Exam tip: when dividing monomials, divide the coefficients and subtract the exponents of like variables.

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

Explanation: Dividing the coefficients gives \(18\div6=3\). When powers with the same base are divided, their exponents are subtracted: \(a^5\div a^2=a^{5-2}=a^3\) and \(b^3\div b=b^{3-1}=b^2\). Hence, the simplified form is \(3a^3b^2\). Option B adds the exponents instead of subtracting them, while option C uses an incorrect coefficient. Exam tip: for division of like bases, subtract the denominator exponent from the numerator exponent.

Correct answer: B. 13

Explanation: Substituting x = 2 gives 3(2³) − 4(2²) + 5 = 3(8) − 4(4) + 5 = 24 − 16 + 5 = 13. Therefore, 13 is correct. Values such as 15 or 17 may result from an error in evaluating the powers or following the order of operations. Exam tip: evaluate powers first, then multiplication, and finally addition or subtraction.

Correct answer: C. 32

Explanation: Substituting \(x=-2\) gives \((-2)^4-2(-2)^3=16-2(-8)=16+16=32\). Therefore, the correct answer is 32. Remember that a negative number has a positive value when raised to an even power, but remains negative when raised to an odd power.

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

Explanation: When dividing powers with the same base, subtract the exponents: \(x^{5-2}=x^3\) and \(y^{-2-(-4)}=y^2\). Therefore, the simplified form is \(x^3y^2\). Option B results from adding the exponents, but division requires their subtraction. Exam tip: Use parentheses carefully when subtracting a negative exponent, as in \(-2-(-4)=2\).

Correct answer: D. \(\frac{x^2}{y^3}\)

Explanation: First, \(\frac{x^{-2}}{y^{-3}}=x^{-2}\times y^3=\frac{y^3}{x^2}\). Raising this expression to the power \(-1\) gives its reciprocal: \(\left(\frac{y^3}{x^2}\right)^{-1}=\frac{x^2}{y^3}\). Option C is only the form inside the outer power, not the final answer. Exam tip: \(a^{-1}\) means \(\frac{1}{a}\).

Correct answer: B. \(\frac{6}{5}\)

Explanation: By the zero-exponent law, the zeroth power of any non-zero number is 1, so \((2^3)^0=1\). By the negative-exponent law, \(5^{-1}=\frac{1}{5}\). Therefore, \(1+\frac{1}{5}=\frac{6}{5}\), so option B is correct. Exam tip: remember that \(a^{-n}=\frac{1}{a^n}\) and \(a^0=1\) for \(a\ne0\).

Correct answer: A. \(\frac{1}{2}\)

Explanation: Use the law of negative exponents: \(a^{-n}=\frac{1}{a^n}\). Thus, \(4^{-1}=\frac{1}{4}\) and \(2^{-2}=\frac{1}{2^2}=\frac{1}{4}\). Therefore, \(\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\), so option A is correct. Exam tip: Convert negative powers into reciprocals before performing the addition.

Correct answer: B. 8

Explanation: Using the law of negative exponents, \(4^{-1}=(2^2)^{-1}=2^{-2}\). Therefore, \(2^5\cdot4^{-1}=2^5\cdot2^{-2}=2^{5-2}=2^3=8\), so option B is correct. Exam tip: when multiplying powers with the same base, add their exponents; when dividing, subtract them.

Correct answer: A. \(4x^2-12x+9\)

Explanation: Apply the identity \((a-b)^2=a^2-2ab+b^2\). Here, \(a=2x\) and \(b=3\), so \((2x-3)^2=(2x)^2-2(2x)(3)+3^2=4x^2-12x+9\). Option B has an incorrect middle term because it does not evaluate \(-2ab\) correctly. Exam tip: in \((a-b)^2\), the middle term is always \(-2ab\).

Correct answer: A. \(9x^2+12x+4\)

Explanation: Use the identity \((a+b)^2=a^2+2ab+b^2\). With \(a=3x\) and \(b=2\), we get \((3x)^2+2(3x)(2)+2^2=9x^2+12x+4\). Therefore, option A is correct. Option B incorrectly uses only \(ab\) instead of the middle term \(2ab\). Exam tip: in \((a+b)^2\), the middle term is always \(2ab\).

Correct answer: A. x⁴

Explanation: By the laws of exponents, powers with the same base are multiplied by adding their exponents and divided by subtracting the exponent in the denominator. Thus, (x³ × x⁵) / x⁴ = x^(3+5−4) = x⁴. Option B results from ignoring the denominator x⁴. Exam tip: Add exponents for multiplication of like bases and subtract them for division.

Correct answer: A. 27

Explanation: For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(\frac{3^4 \times 3^2}{3^3}=3^{4+2-3}=3^3=27\). Exam tip: apply the exponent laws before evaluating the final power.

Correct answer: B. 3

Explanation: Since \(81=3^4\), the equation becomes \(3^{x+1}=3^4\). Because the bases are equal and positive, their exponents must be equal: \(x+1=4\), so \(x=3\). Option C is a common error because substituting \(x=4\) gives an exponent of \(5\), not \(4\). Exam tip: Express both sides with the same base before equating the exponents.