Exponents

Rewrite 9 as 3^2: 9^3=(3^2)^3=3^6. Using the quotient rule for powers with the same base, 3^6 \div 3^4=3^{6-4}=3^2=9. Therefore, option A is correct. Exam tip: when dividing powers with the same base, subtract the exponents.

Using the rule for equal exponents, \(2^3 \times 5^3=(2\times5)^3=10^3\). Therefore, \(10^3\div10^2=10^{3-2}=10\), so option B is correct. Option A results from an incorrect evaluation of the powers. Exam tip: when dividing powers with the same base, subtract their exponents.

In \((a^m)^n\), the exponent \(m\) is repeated through the outer power \(n\), so the total exponent is \(m\times n=mn\). The \(m+n\) rule applies only to \(a^m\times a^n\). Exam tip: check brackets first.

Express each number as a power of 2: 8^2=(2^3)^2=2^6 and 4^2=(2^2)^2=2^4. Therefore, 2^6 \times 2^3 \div 2^4=2^{6+3-4}=2^5=32. Hence, option D is correct. Exam tip: when powers have the same base, add exponents while multiplying and subtract them while dividing.

Given x=2^4, so x^2=(2^4)^2=2^8. Therefore, x^2\div2^5=2^8\div2^5=2^{8-5}=2^3=8. For division with the same base, subtract the exponents; hence 16 is not correct. Exam tip: remember the rule a^m\div a^n=a^{m-n}.

For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(6^3 \times 6^2 \div 6^4=6^{3+2-4}=6^1=6\). Therefore, option B is correct. Remember that the exponent is subtracted when dividing powers with the same base, not added.

First apply the power-of-a-power rule: \((2^3)^2=2^{3\times2}=2^6\). Therefore, \(2^{10}\div2^6=2^{10-6}=2^4=16\), so option D is correct. Exam tip: when dividing powers with the same base, subtract the exponents; for a power raised to another power, multiply the exponents.

Write 125 as a power of 5: 125 = 5^3. Therefore, 125^2 \div 5^4 = (5^3)^2 \div 5^4 = 5^6 \div 5^4 = 5^{6-4} = 5^2 = 25. Hence, 25 is correct. Exam tip: when dividing powers with the same base, subtract the exponents. Option 125 would result from using an incorrect exponent difference.

Using the exponent rule ((a^m)^n=a^{mn}), (4^3)^2=4^6. Also, 16^2=(4^2)^2=4^4. Therefore, 4^6 ÷ 4^4=4^2=16, so option B is correct. Exam tip: when dividing powers with the same base, subtract their exponents.

First evaluate the two powers: 3^5 = 243 and 2^7 = 128. Therefore, 3^5 + 2^7 = 243 + 128 = 371, so option C is correct. In such questions, calculate each power separately before performing the addition.

For division of powers with the same base, use the rule \(a^m\div a^n=a^{m-n}\). Thus, \(2^{12}\div2^8=2^{12-8}=2^4=16\). Therefore, 16 is correct. Exam tip: when dividing powers with the same base, subtract the exponents; do not add them as in multiplication.

When powers with the same base are divided, their exponents are subtracted: \(10^4\div10^2=10^{4-2}=10^2=100\). Therefore, the correct answer is 100. The value 10 would result from reducing the exponent by only one, so it is incorrect. Exam tip: For division with the same base, use \(a^m\div a^n=a^{m-n}\).

For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(2^5\times2^4\div2^6=2^{5+4-6}=2^3=8\). Therefore, option B is correct. Exam tip: When the base is the same, apply the exponent laws first and then evaluate the resulting power.

First evaluate the powers: 9^2 = 81 and 5^2 = 25. Therefore, 9^2 - 5^2 = 81 - 25 = 56, so 56 is correct. Exam tip: calculate both squares before subtracting; alternatively, use the difference of squares: (9 - 5)(9 + 5) = 4 × 14 = 56.

Rewrite the numbers with the same base: \(8=2^3\), so \(8^2=(2^3)^2=2^6\). Therefore, \(2^6\div2^4=2^{6-4}=2^2=4\). Hence, the correct answer is 4. Exam tip: when dividing powers with the same base, subtract their exponents.

Using the power-of-a-power rule, \((3^3)^2=3^{3\times2}=3^6\). When powers with the same base are divided, their exponents are subtracted: \(3^6\div3^4=3^{6-4}=3^2=9\). Therefore, 9 is correct. Exam tip: Apply \((a^m)^n=a^{mn}\) first, followed by \(a^m\div a^n=a^{m-n}\).

For division of powers with the same base, subtract the exponents: \(6^3\div6^1=6^{3-1}=6^2=36\). Therefore, option C is correct. Option A uses only the base, while option B gives the value of \(6^3\) without carrying out the division. Exam tip: remember the rule \(a^m\div a^n=a^{m-n}\) for dividing powers with the same base.

Since \(25=5^2\), we get \(25^2=(5^2)^2=5^4\). Therefore, \(25^2\div5^3=5^4\div5^3=5^{4-3}=5\). Hence, option B is correct. Exam tip: When dividing powers with the same base, subtract the exponents.

Evaluate the powers separately first: 11^2 = 121 and 4^3 = 64. Therefore, 11^2 + 4^3 = 121 + 64 = 185, so option B is correct. In an exam, calculate each exponent before performing the addition.

Write 32 as a power of 2: 32 = 2^5. Therefore, 2^{10} ÷ 32 = 2^{10} ÷ 2^5 = 2^{10-5} = 2^5 = 32. Hence, option C is correct. Exam tip: When dividing powers with the same base, subtract their exponents.