Operations on real numbers and the laws of exponents

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

Explanation: Use the identity \((a+b)^2=a^2+2ab+b^2\). Taking \(a=x\) and \(b=9\), we get \((x+9)^2=x^2+2\cdot x\cdot9+9^2=x^2+18x+81\). Therefore, option A is correct. Option B would give the middle term \(-18x\), while option D omits the \(18x\) term. Exam tip: Take the square root of the constant term and use the sign of the middle term to choose between \((x+a)^2\) and \((x-a)^2\).

Correct answer: A. (x-10)^2

Explanation: Using the perfect-square identity (a-b)^2=a^2-2ab+b^2, we get x^2-20x+100=x^2-2\cdot x\cdot10+10^2=(x-10)^2. Option B would produce a middle term of +20x, so it is incorrect. Exam tip: take the square roots of the first and last terms and check whether twice their product matches the middle term.

Correct answer: A. 73

Explanation: Use the identity \(m^2+n^2=(m+n)^2-2mn\). Thus, \(m^2+n^2=11^2-2(24)=121-48=73\). Option B results from subtracting \(mn\) only once, whereas the identity requires subtracting \(2mn\). In an exam, writing the relevant algebraic identity first is the quickest method.

Correct answer: C. 69

Explanation: Use the identity \(a^2+b^2=(a-b)^2+2ab\). Thus, \(a^2+b^2=7^2+2(10)=49+20=69\). Therefore, the correct answer is 69. Remember not to omit the term \(2ab\); using only \((a-b)^2+ab\) would incorrectly give 59.

Correct answer: A. 9604

Explanation: Since \(98=100-2\), use the identity \((a-b)^2=a^2-2ab+b^2\): \(98^2=(100-2)^2=100^2-2(100)(2)+2^2=10000-400+4=9604\). Therefore, option A is correct. Exam tip: For squaring a number close to a convenient base such as 100, use the identity \((a\pm b)^2\).

Correct answer: A. 10609

Explanation:
(103)^2=(100+3)^2=100^2+2(100)(3)+3^2=10000+600+9=10609
, so option A is correct. In exams, remember the middle term
(2ab)
in
(a+b)^2=a^2+2ab+b^2
; omitting it can lead to the incorrect answer 10009.

Correct answer: A. 3584

Explanation: 64 × 56 = (60 + 4)(60 − 4). Using the identity (a + b)(a − b) = a² − b², we get 60² − 4² = 3600 − 16 = 3584. Therefore, 3584 is correct. Exam tip: Use the difference-of-squares identity when two factors are equally spaced around the same middle number.

Correct answer: B. (41)

Explanation: The direct answer is B, 41. Use the negative-exponent rule: for any nonzero number a, a^{-n}=1/a^n, so a fraction raised to a negative power is inverted before the positive power is taken. Thus \((1/4)^{-2}=4^2=16\), and \((1/5)^{-2}=5^2=25\). Adding gives \(16+25=41\). Option A, 9, is wrong because neither the correct powers nor their sum gives 9. Option B, 41, is correct because it is exactly 16+25. Option C, \(41/400\), incorrectly treats the negative powers as if the fractions remained in the numerator; it is not the value of the sum. Option D, 400, is also incorrect: 400 would come from an unrelated multiplication, not this addition. Remember: a negative exponent flips the fraction, and then add the resulting values.

Correct answer: C. 20

Explanation: Substituting the given values, \(a^2b+ab^2=4^2\times1+4 imes1^2=16+4=20\). Alternatively, factorising gives \(ab(a+b)=4 imes1 imes(4+1)=20\). In an exam, evaluate the powers before carrying out the multiplication.

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

Explanation: The greatest common factor of \(3x^4\) and \(9x^3\) is \(3x^3\). Taking it outside gives \(3x^4+9x^3=3x^3(x+3)\), since \(3x^3\times x=3x^4\) and \(3x^3 imes3=9x^3\). Option C is not a valid factorisation because factoring out 9 would require a fractional coefficient in the bracket, not \(x+1\). Exam tip: multiply the factor outside the bracket by every term inside to verify the result.

Correct answer: A. \(4a^3b\)

Explanation: Dividing the numerical coefficients gives \(24\div6=4\). For powers with the same base, subtract the exponents: \(a^{5-2}=a^3\) and \(b^{3-2}=b\). Hence, \(\frac{24a^5b^3}{6a^2b^2}=4a^3b\), so option A is correct. In option B, the exponent of \(b\) has not been reduced correctly. Exam tip: for division of like bases, use \(x^m\div x^n=x^{m-n}\).

Correct answer: A. \(10x^4y^3\)

Explanation: Multiply the numerical coefficients to get \(5\times2=10\). For like bases, add the exponents: \(x^{3+1}=x^4\) and \(y^{1+2}=y^3\). Therefore, the product is \(10x^4y^3\). Option C omits one power of \(x\), while option D combines the exponents incorrectly. Exam tip: when multiplying monomials, multiply the coefficients and add the exponents of like variables.

Correct answer: C. 13

Explanation: Substituting \(x=3\) gives \(x^3=27\) and \(x^2=9\). Therefore, \(2x^3-5x^2+4=2(27)-5(9)+4=54-45+4=13\), so option C is correct. As an exam tip, evaluate powers first and then perform multiplication, addition, and subtraction in the proper order.

Correct answer: B. 54

Explanation: \\((-3)^4=81\\) because an even power gives a positive result, while \\((-3)^3=-27\\) because an odd power retains the negative sign. Therefore, \\(x^4+x^3=81+(-27)=54\\), so option B is correct. Exam tip: Always check the sign when raising a negative number to an even or odd power.

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

Explanation: For division with like bases, subtract the exponent in the denominator from the exponent in the numerator: \(x^{6-3}y^{-3-(-5)}=x^3y^2\). Therefore, \(x^3y^2\) is correct. Option B mishandles the exponents of \(y\), while C and D incorrectly add the exponents of \(x\). Exam tip: when dividing powers with the same base, subtract the denominator exponent from the numerator exponent.

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

Explanation: First, \(\frac{x^{-3}}{y^{-2}}=x^{-3}\times y^2=\frac{y^2}{x^3}\). Applying the outer power \(-1\) takes the reciprocal: \(\left(\frac{y^2}{x^3}\right)^{-1}=\frac{x^3}{y^2}\). Therefore, option A is correct. Option B is only the simplified inner expression and does not include the effect of the outer \(-1\) power. Exam tip: the \(-1\) power of any nonzero expression gives its reciprocal.

Correct answer: B. \(\frac{5}{4}\)

Explanation: Any non-zero number raised to the power 0 equals 1, so \((3^2)^0=1\). Using the negative-exponent rule \(a^{-n}=\frac{1}{a^n}\), we get \(4^{-1}=\frac{1}{4}\). Therefore, \(1+\frac{1}{4}=\frac{5}{4}\), so option B is correct. Option A incorrectly omits the contribution of the first term. Exam tip: apply the zero-exponent and negative-exponent rules separately before adding.

Correct answer: B. \(\frac{2}{9}\)

Explanation: Using the negative exponent rule \(a^{-n}=\frac{1}{a^n}\), we get \(9^{-1}=\frac{1}{9}\) and \(3^{-2}=\frac{1}{3^2}=\frac{1}{9}\). Therefore, the sum is \(\frac{1}{9}+\frac{1}{9}=\frac{2}{9}\). A useful exam tip is to convert negative powers into reciprocals before performing the operation.

Correct answer: B. 27

Explanation: Since \(9=3^2\), we have \(9^{-1}=(3^2)^{-1}=3^{-2}\). Therefore, \(3^5\cdot 9^{-1}=3^5\cdot 3^{-2}=3^{5-2}=3^3=27\), so option B is correct. Exam tip: When multiplying powers with the same base, add their exponents.

Correct answer: A. \(9x^2-24x+16\)

Explanation: Apply the identity \((a-b)^2=a^2-2ab+b^2\). Here, \(a=3x\) and \(b=4\). Thus, \((3x-4)^2=(3x)^2-2(3x)(4)+4^2=9x^2-24x+16\). Therefore, option A is correct. Remember that the middle term is \(-2ab\), not just \(-ab\).

Correct answer: A. \(a>0\)

Explanation: For real exponents, \(a^m\) is generally defined for \(a>0\), so the product law applies safely. With \(a<0\), some irrational powers are not real. Exam tip: whenever real exponents appear, first check that the base is positive.

Correct answer: A. \(36x^2-25\)

Explanation: Use the identity \((a+b)(a-b)=a^2-b^2\). Here, \(a=6x\) and \(b=5\), so \((6x)^2-5^2=36x^2-25\). Therefore, option A is correct. Option B incorrectly uses the sum of the squares, while the given factors form a difference of squares. Exam tip: whenever two binomials differ only in the sign between the terms, check the identity \((a+b)(a-b)\).

Correct answer: B. 5

Explanation: Since \(64=2^6\), the equation becomes \(2^{x+1}=2^6\), so \(x+1=6\). Therefore, \(x=5\). Choosing 6 is a common mistake because it is the value of \(x+1\), not \(x\). In the exam, first express both sides with the same base and then equate the exponents.

Correct answer: C. 5

Explanation: Since \(125=5^3\), the equation becomes \(5^{x-2}=5^3\). With equal bases, the exponents are equal, so \(x-2=3\), giving \(x=5\). Exam tip: rewrite the numerical side as a power of the same base before equating exponents.

Correct answer: A. 2

Explanation: The governing concept is the law of exponents: \((a^m)^n=a^{mn}\), negative exponents represent reciprocals, and division by a power subtracts its exponent. Rewrite every quantity with base 2. We have \((2^3)^2=2^6\), \(4^{-1}=(2^2)^{-1}=2^{-2}\), and \(8=2^3\). Therefore the expression is \(2^6\cdot2^{-2}\div2^3=2^{6-2-3}=2^1=2\). Hence option A is correct. A common error is to treat \(4^{-1}\) as 4 or to add the denominator exponent instead of subtracting it. Options B, C and D reflect such exponent or reciprocal mistakes.