Operations on real numbers and the laws of exponents

Correct answer: C. 10

Explanation: The governing concept is the quotient law of exponents: for a non-zero common base, x^a/x^b=x^(a−b). Comparing the first condition with x^7 gives a−b=7. The second condition supplies a+b=13. Adding these two linear equations eliminates b: (a−b)+(a+b)=7+13, so 2a=20 and a=10. A check gives b=3; then a−b=10−3=7 and a+b=10+3=13, confirming both original conditions. Therefore option C is correct. Option A is actually the resulting value of b, option B is the exponent difference rather than a, and option D is the sum a+b, not the requested individual exponent.

Correct answer: C. 5

Explanation: The governing concept is converting every number to a power of the same base and then applying exponent laws. Since 100=10², we obtain 100²=(10²)²=10⁴. Also, 1000=10³. Therefore the left-hand side becomes 10^m×10⁴/10³=10^(m+4−3)=10^(m+1). The equation is consequently 10^(m+1)=10⁶. Because the base 10 is positive and different from 1, equal powers have equal exponents, so m+1=6 and m=5. Hence option C is correct. The other choices can result from forgetting to square 100, adding rather than subtracting the denominator exponent, or simply copying the final exponent without simplifying the other factors.

Correct answer: B. x^3+8

Explanation: The governing concept is the difference-of-squares identity, A^2−B^2=(A−B)(A+B). Rewrite the numerator as x^6−64=(x^3)^2−8^2. It therefore factors as (x^3−8)(x^3+8). Since the condition x^3≠8 guarantees that the denominator is non-zero, the common factor x^3−8 may be cancelled, leaving x^3+8. Thus option B is correct. Option A is only the cancelled denominator factor, option C does not follow from a valid factorization, and option D incorrectly adds the two square terms instead of applying the difference identity. The restriction remains important even after simplification because the original expression was undefined when x^3=8.

Correct answer: A. -7

Explanation: The governing concept is expressing exponential equations with a common base. Since 32=2^5, the equation 2^a=32 gives 2^a=2^5 and therefore a=5. Also, 8=2^3 and 64=2^6. Thus 8^b=(2^3)^b=2^(3b)=2^6, so 3b=6 and b=2. Substituting into the required expression gives a^b−b^a=5^2−2^5=25−32=−7. Hence option A is correct. Option B would result from reversing the subtraction order. Options C and D do not follow from a=5 and b=2 and indicate an incorrect interpretation of the powers or the given equations. The common-base method avoids such confusion.

Correct answer: C. 5

Explanation: The governing concept is the laws of exponents after expressing all numerical factors with base 10. Since 1000=10³, its square is 1000²=(10³)²=10⁶. Also, 100=10². Therefore the left side becomes (10^k×10⁶)/10²=10^(k+6−2)=10^(k+4). The equation is then 10^(k+4)=10⁹. Equal powers of the same valid base 10 have equal exponents, so k+4=9 and k=5. Thus option C is correct. The alternatives can arise from forgetting the square on 1000, mishandling division by 100, or treating the final exponent 9 as the answer without accounting for the other factors. Substitution gives 10⁵×10⁶/10²=10⁹, confirming the result.

Correct answer: C. 5

Explanation: The governing concept is equality of powers after rewriting all numbers with a common base. Use base 3: 9=3^2, 27=3^3, and 729=3^6. From 9^x=729, we obtain (3^2)^x=3^(2x)=3^6, so 2x=6 and x=3. From 27^y=729, we obtain (3^3)^y=3^(3y)=3^6, so 3y=6 and y=2. Consequently x+y=3+2=5, which is option C. The fractional choices arise from confusing the original bases with their exponents or dividing incorrectly. Option D does not represent the sum of the correctly determined exponents. Rewriting both equations with base 3 makes the comparison unambiguous.

Correct answer: B. 2

Explanation: The governing concept is conversion to a common base and then comparison of exponents. Since 27 = 3^3, we have 27^(x−1) = (3^3)^(x−1) = 3^(3x−3). Therefore the left-hand side becomes 3^x · 3^(3x−3) = 3^(4x−3). Also, 243 = 3^5. The bases are equal and the base 3 is positive and different from 1, so their exponents must be equal: 4x − 3 = 5. Adding 3 gives 4x = 8, and dividing by 4 gives x = 2. Substitution confirms the result: 3^2 · 27^1 = 9 · 27 = 243. Hence option B is correct. Option A leaves the exponent too small, while C and D do not satisfy the original equation.

Correct answer: C. 5

Explanation: The governing concept is expressing every quantity as a power of the same base, 10. We have 100 = 10^2, so 100^3 = (10^2)^3 = 10^6. Similarly, 1000 = 10^3, so 1000^2 = (10^3)^2 = 10^6. Substituting these values gives (10^k · 10^6)/10^6 = 10^k, because equal powers in the numerator and denominator cancel. The equation therefore reduces to 10^k = 10^5. Since the base 10 is positive and not equal to 1, equal powers have equal exponents, so k = 5. Thus option C is correct. The values 3, 4, and 6 would produce 10^3, 10^4, and 10^6 respectively after simplification, so none of them can equal 10^5.