Operations on real numbers and the laws of exponents

Correct answer: B. 2

Explanation: 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.

Correct answer: C. 4

Explanation: The governing algebraic concept is the difference-of-squares identity, \(a^2-b^2=(a-b)(a+b)\). Taking \(a=2.5\) and \(b=1.5\), the expression becomes \((2.5-1.5)(2.5+1.5)\). The first factor is \(1\), and the second factor is \(4\), so their product is \(1\times4=4\). Direct checking gives \(2.5^2=6.25\) and \(1.5^2=2.25\), whose difference is also \(6.25-2.25=4\). Thus option C is correct. Options A, B and D can result from an arithmetic error or from applying the identity incorrectly; the decimal values do not change the validity of the identity.

Correct answer: B. 6

Explanation: Use the difference-of-squares identity \(a^2-b^2=(a-b)(a+b)\), which is valid for real numbers, including decimals. Here \(a=3.5\) and \(b=2.5\). Thus \(3.5^2-2.5^2=(3.5-2.5)(3.5+2.5)=1\times6=6\). A direct check gives \(3.5^2=12.25\) and \(2.5^2=6.25\), and \(12.25-6.25=6\), confirming the result. Therefore option B is correct. Option A confuses the difference with one of the factors, while options C and D arise from incorrect addition or squaring. Factoring is especially efficient here because the two numbers differ by exactly 1.

Correct answer: A. 4

Explanation: The governing concept is equality of powers with the same positive base. Express 32 as a power of 2: \(32=2^5\). The equation therefore becomes \(2^{x+1}=2^5\). Since the base 2 is the same on both sides and is not 0 or 1, the exponents must be equal, giving \(x+1=5\). Subtracting 1 from both sides yields \(x=4\). Substitution confirms the result: \(2^{4+1}=2^5=32\). Thus option A is correct. Option B gives the exponent \(x+1\), not \(x\); option C results from subtracting incorrectly, and option D adds rather than subtracts 1.

Correct answer: A. 27

Explanation: The governing concept is the law of exponents, especially (a^m)^n = (a^n)^m and the ability to rewrite a number using a known base. Since 8 = 2^3, we can write 8^x = (2^3)^x = (2^x)^3. The given relation is 2^x = 3, so substitution gives 8^x = 3^3 = 27. Hence option A is correct. The answer 9 would incorrectly use a square instead of a cube. The value 24 does not follow from any applicable exponent law, and 6 incorrectly multiplies the known value by 2 rather than raising it to the third power. The calculation uses only the stated relation and valid power rules.

Correct answer: A. -15

Explanation: The governing concept is direct evaluation of a polynomial by substitution. Replace every x with −1 and preserve brackets so the powers and signs are interpreted correctly: P(−1)=2(−1)^3−5(−1)^2+(−1)−7. Since (−1)^3=−1 and (−1)^2=1, this becomes −2−5−1−7. Adding the terms gives −2−5=−7, then −7−1=−8, and finally −8−7=−15. Therefore option A is correct. Option B can result from omitting a term or making a sign error. Option C reverses the final sign, and option D is merely one input-related value rather than the value of the whole polynomial. Brackets are especially important when substituting a negative number.

Correct answer: A. (uv)^n=u^n v^n

Explanation: The governing law is the power-of-a-product rule: when a product is raised to a common exponent n, the exponent applies separately to each factor. Thus (uv)^n=u^n v^n. For example, with u=2, v=3, and n=2, the left side is (2·3)^2=6^2=36, while the right side is 2^2·3^2=4·9=36. Therefore option A is correct. Option B wrongly changes multiplication into addition. Option C reverses the product rule and is not generally true; for instance, 2^2·3^2=36 but (2+3)^2=25. Option D combines different exponents without a valid exponent law; multiplication of powers with the same base, not different bases, is where exponents are added.