Exponents

Given \(x=\sqrt{11}\), we get \(x^2=(\sqrt{11})^2=11\). Therefore, \(x^2+5=11+5=16\), so option B is correct. \(121\) would result from squaring 11 itself, but here \(\sqrt{11}\) is being squared. Exam tip: squaring a square root gives the original non-negative number.

Given \(x=\sqrt{3}\), we get \(x^2=(\sqrt{3})^2=3\). Therefore, \(x^2+x^2=3+3=6\), so 6 is correct. Option 9 would result from calculating \(3^2\), but the value of \(x^2\) is 3. Exam tip: squaring a square root gives the original non-negative number.

Given \(x=\sqrt{2}\), we get \(x^2=(\sqrt{2})^2=2\). Squaring a principal square root gives the number inside the root. Option 1 would apply only if \(x=1\); here the square of \(\sqrt{2}\) is 2. Exam tip: apply \((\sqrt{a})^2=a\) directly.

For powers with the same base, exponents are added during multiplication and subtracted during division: \(2^8 \times 2^5 \div 2^9=2^{8+5-9}=2^4=16\). Therefore, the correct answer is 16. The option 8 equals only \(2^3\), so it is incorrect. Exam tip: for the same base, add exponents when multiplying and subtract them when dividing.

First apply the power-of-a-power rule: (a^m)^n = a^{mn}, so (3^2)^4 = 3^8. Then use the quotient rule for powers with the same base: 3^8 \div 3^5 = 3^{8-5} = 3^3 = 27. Therefore, option B is correct. Exam tip: multiply exponents for a power raised to a power, but subtract exponents when dividing like bases.

A negative exponent represents a reciprocal. Thus, \((ab)^{-n}=\frac{1}{(ab)^n}=\frac{1}{a^n b^n}=a^{-n}b^{-n}\). Option B incorrectly uses addition; there is no exponent law that turns the reciprocal of a product into a sum of reciprocals. Exam tip: When an exponent is applied to a product in brackets, first apply the power-of-a-product rule.

A negative exponent does not make the base or the result negative; it denotes the reciprocal of the corresponding positive power. Therefore, \(a^{-n}=\dfrac{1}{a^n}\) for \(a\ne0\). Option B wrongly interprets the negative exponent as a minus sign, while option D is true only in some special cases, not generally. Exam tip: Rewrite every negative exponent as a reciprocal before simplifying.

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.