Operations on real numbers and the laws of exponents

\(53\cdot47=(50+3)(50-3)\). 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.

The governing concept is the difference-of-squares identity, a^2-b^2=(a-b)(a+b). Apply it with a=1.5 and b=0.5: (1.5)^2-(0.5)^2=(1.5-0.5)(1.5+0.5). The first factor is 1 and the second factor is 2, so their product is 1×2=2. Therefore option B is correct. Direct checking gives 1.5^2=2.25 and 0.5^2=0.25, whose difference is also 2. Option A may result from calculating only the difference of the numbers; option C can come from adding incorrectly, and option D reflects an arithmetic error. The identity works equally well with decimal numbers, provided the signs and parentheses are handled carefully.

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.

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.

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.

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.

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.

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.

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.

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 \(-2-(-4)=2\).

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 \(\frac{1}{a}\).

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 \(a\ne0\).

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.

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.

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 \(-2ab\).

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 \(2ab\).

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.

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.

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.

A negative exponent represents a reciprocal. Thus, \((ab)^{-1}=\frac{1}{ab}=\frac{1}{a}\cdot\frac{1}{b}=a^{-1}b^{-1}\). No similar rule applies to a sum or difference. Exam tip: distinguish a product from a sum before using exponent laws.