Exponents

Using the product rule for equal exponents, \(a^n \times b^n=(ab)^n\). Thus, \(2^3 \times 5^3=(2\times5)^3=10^3=1000\), so option B is correct. Exam tip: When the exponents are equal, multiply the bases first and then apply the common exponent.

First evaluate the powers: 7^2 = 49 and 2^5 = 32. Adding them gives 49 + 32 = 81, so option B is correct. Exam tip: evaluate exponents before performing the addition.

Evaluate the powers first: \(13^2=169\) and \(5^2=25\). Therefore, \(13^2-5^2=169-25=144\), so option A is correct. Exam tip: calculate both squares before subtracting; option B is only the value of \(13^2\), not of the complete expression.

Since \(a^n\times a^{-n}=a^{n-n}=a^0=1\), \(a^{-n}\) is the reciprocal of \(a^n\). Hence it equals \(\frac{1}{a^n}\), not \(-a^n\). Exam tip: convert a negative exponent into a reciprocal first.

Using the law \(\frac{a^n}{b^n}=\left(\frac{a}{b}\right)^n\) for equal exponents, \(\frac{15^2}{5^2}=\left(\frac{15}{5}\right)^2=3^2=9\). Therefore, 9 is correct. Exam tip: When the exponents are the same, divide the bases first and then apply the exponent.

For powers with the same base, the exponents are added: \(2^3\times2^2=2^{3+2}=2^5\). Then \((2^5)^2=2^{5\times2}=2^{10}=1024\). Therefore, option B is correct. Option C, 512, equals \(2^9\) and may result from applying the outer exponent incorrectly. Exam tip: remember \(a^m\times a^n=a^{m+n}\) and \((a^m)^n=a^{mn}\).

First evaluate the two powers: \(3^5=243\) and \(2^5=32\). Therefore, \(3^5+2^5=243+32=275\), so option D is correct. Option B is only the value of \(3^5\), not the complete expression. Exam tip: Evaluate each power separately before performing the addition.

Using the quotient rule for equal powers, \((25^2 \div 5^2)=(25\div 5)^2=5^2=25\). Therefore, option B is correct. Option A is only the quotient of the bases and misses the required squaring step. In an exam, remember the rule \(a^n\div b^n=(a\div b)^n\) when the exponents are equal.

Evaluate the powers first: 2^4=16 and 4^3=64. Therefore, 2^4+4^3=16+64=80, so option C is correct. In an exam, remember to calculate each power before performing the addition; 72 may result from an error in evaluating the powers.

First calculate the two squares: 14^2 = 196 and 10^2 = 100. Therefore, 14^2 - 10^2 = 196 - 100 = 96. This can also be verified using the difference-of-squares identity: a^2 - b^2 = (a-b)(a+b) = (14-10)(14+10) = 4 × 24 = 96. Exam tip: Use the difference-of-squares identity to simplify such expressions quickly.

Using the quotient rule for equal exponents, \(a^n \div b^n=(a\div b)^n\). Therefore, \(8^3\div2^3=(8\div2)^3=4^3=64\). Hence, 64 is correct. Exam tip: do not confuse \(8^3\div2^3\) with \(8\div2^3\); the placement of the exponent changes the expression.

First evaluate the powers: 2^7 = 128 and 2^4 = 16. Therefore, 2^7 - 2^4 = 128 - 16 = 112, so option D is correct. Remember that the exponent subtraction rule cannot be applied directly here because the terms are being subtracted; evaluate the two powers first.

Using the exponent rule \((a^m)^n=a^{mn}\), \((5^3)^2=5^{3\times2}=5^6=15625\). Therefore, 15625 is the correct answer. 3125 equals \(5^5\) and may result from handling the exponents incorrectly. Exam tip: when a power is raised to another power, multiply the exponents.

For division of numbers with the same exponent, use
\((a^n \div b^n)=(a\div b)^n\). Thus,
\((16^2 \div 4^2)=(16\div4)^2=4^2=16\). Therefore, option C is correct. Exam tip: when the exponents are equal, divide the bases first and then apply the common exponent.

First evaluate the powers: 11^2 = 121 and 2^6 = 64. Therefore, 11^2 + 2^6 = 121 + 64 = 185, so option C is correct. Exam tip: Evaluate each exponent separately before performing the addition to avoid calculation errors.

First evaluate the powers: 12^2=144 and 5^2=25. Therefore, 12^2+5^2=144+25=169, so option A is correct. In an exam, calculate each square separately before adding; 179 represents a likely arithmetic error in the addition.

Evaluate the powers first: \(10^3=1000\) and \(9^2=81\). Therefore, \(10^3-9^2=1000-81=919\), so option B is correct. Exam tip: calculate the powers before performing the subtraction.

First evaluate the powers: 7^3 = 7 × 7 × 7 = 343 and 6^2 = 6 × 6 = 36. Therefore, 7^3 + 6^2 = 343 + 36 = 379, so option C is correct. In an exam, calculate each power separately before adding to avoid errors.

13^2 = 13 × 13 = 169 and 3^4 = 3 × 3 × 3 × 3 = 81. Therefore (13^2 + 3^4) = 169 + 81 = 250. Option A (240) is incorrect — it likely arises from an arithmetic mistake (for example, computing 3^4 incorrectly). Exam tip: evaluate each power separately, then add and recheck the addition to avoid simple errors.

When powers with the same base are divided, their exponents are subtracted: \(2^9 \div 2^6=2^{9-6}=2^3=8\). Therefore, option B is correct. Exam tip: for division with the same base, subtract the exponents rather than adding them.