Exponents

For division of powers with the same base, subtract the exponents: \(\frac{2^{x+3}}{2^{x-1}}=2^{(x+3)-(x-1)}=2^4=16\). Thus, the given equation is true for every real value of \(x\), so there is no single fixed value of \(x\). Therefore, option C is correct. Exam tip: use \(a^m/a^n=a^{m-n}\) for powers with the same non-zero base; here, \(x\) cancels out.

Using the fractional exponent rule \(a^{m/n}=(\sqrt[n]{a})^m\), we get \(\left(\frac{27}{8}\right)^{2/3}=\left(\sqrt[3]{\frac{27}{8}}\right)^2=\left(\frac{3}{2}\right)^2=\frac{9}{4}\). Therefore, option A is correct. Option B is only the cube root, \(\frac{3}{2}\), and does not include the required square. Exam tip: for an exponent \(m/n\), take the \(n\)th root and then raise the result to the power \(m\).

Express 4 and 64 as powers of 2: \(4^{x+1}=2^{2x+2}\) and \(64^{x-1}=2^{6x-6}\). Since the bases are equal, equate the exponents: \(2x+2=6x-6\). Thus, \(8=4x\), giving \(x=2\). Exam tip: When an exponential equation is reduced to the same base, equate the exponents.

When powers with the same base are multiplied, their exponents are added: \(a^m \times a^n = a^{m+n}\). Thus, \(a^3 \times a^4 = a^{3+4}=a^7\), so option A is correct. Option B incorrectly multiplies the exponents, which is not the rule for multiplication of like bases. Exam tip: add exponents for multiplication of like bases and subtract them for division.

When powers with the same base are multiplied, their exponents are added: x^5 × x^2 = x^{5+2} = x^7. Therefore, option A is correct. Option B incorrectly multiplies the exponents, whereas this rule requires their addition. Exam tip: for multiplication of like bases, keep the base unchanged and add the exponents.

For a power raised to another power, use the rule \((a^m)^n=a^{mn}\). Therefore, \((a^2)^3=a^{2\times3}=a^6\), so option A is correct. Option C is incorrect because it adds the exponents to get 9; exponents are multiplied in this situation. Exam tip: When a power is raised to another power, multiply the exponents.

When powers with the same base are multiplied, their exponents are added: 2^3 × 2^4 = 2^(3+4) = 2^7. Therefore, option A is correct. Option B incorrectly multiplies the exponents, which is not the rule for multiplying like bases. Exam tip: remember a^m × a^n = a^(m+n) for multiplication of powers with the same base.

When powers with the same non-zero base are divided, their exponents are subtracted: \(\frac{a^m}{a^n}=a^{m-n}\), where \(a\ne0\). Thus, \(\frac{a^6}{a^2}=a^{6-2}=a^4\). Exam tip: subtract exponents for division of like bases; do not add them.

For division of powers with the same non-zero base, subtract the exponents: \(\frac{x^m}{x^n}=x^{m-n}\), where \(x\neq0\). Thus, \(\frac{x^8}{x^5}=x^{8-5}=x^3\). Option B incorrectly adds the exponents, which is the rule for multiplication, not division. Exam tip: add exponents in multiplication and subtract them in division when the bases are the same.

For any non-zero number or variable, the exponent rule gives \(a^m \div a^m=a^{m-m}=a^0\). The left-hand side equals 1, so \(a^0=1\) for \(a\neq 0\). Therefore, option A is correct. Option C would apply when the exponent is 1, not 0. Exam tip: the zeroth power of every non-zero base is 1.

When a power is raised to another power, the exponents are multiplied: \((x^3)^2=x^{3\times2}=x^6\). Therefore, option A is correct. Remember that the exponents are multiplied, not added, so \(x^5\) is incorrect.

Use the power-of-a-power rule: \((a^m)^n=a^{mn}\). Therefore, \((2^2)^3=2^{2\times3}=2^6=64\), so option A is correct. Option C is incorrect because the exponents are not combined by adding or arbitrarily increasing them; they must be multiplied. Exam tip: when a power is raised to another power, multiply the exponents.

When powers with the same base are multiplied, their exponents are added: 3^4 × 3^1 = 3^(4+1) = 3^5. Therefore, option A is correct. Remember: add exponents for multiplication with the same base, but subtract them for division.

For division of powers with the same base, use \(\frac{a^m}{a^n}=a^{m-n}\). Thus, \(\frac{2^7}{2^3}=2^{7-3}=2^4\), so option A is correct. Option B results from adding the exponents, which is the rule for multiplication, not division. Exam tip: subtract the exponents when dividing powers with the same non-zero base.

The rule for a negative exponent is \(a^{-n}=\frac{1}{a^n}\), where \(a\neq0\). Therefore, \(a^{-2}=\frac{1}{a^2}\). Option B is the positive exponent form, while option D represents only \(a^{-1}\). Exam tip: for a negative exponent, take the reciprocal of the base and change the exponent to positive.

The negative-exponent rule is \(x^{-n}=\frac{1}{x^n}\), where \(x\neq0\). Therefore, \(x^{-3}=\frac{1}{x^3}\). Option B represents a positive exponent, while option C incorrectly adds a minus sign instead of taking the reciprocal. Exam tip: for a negative exponent, take the reciprocal of the base and make the exponent positive.

For \(a\neq 0\), the negative-exponent rule is \(a^{-n}=\frac{1}{a^n}\). Thus, \(a^{-4}=\frac{1}{a^4}\), so \(\frac{1}{a^{-4}}=\frac{1}{1/a^4}=a^4\). Therefore, option A is correct. Exam tip: first rewrite a negative exponent as a reciprocal, then simplify the resulting fraction.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((a^{-1})^3=a^{(-1)\times3}=a^{-3}\), provided \(a\ne0\). Option B ignores the negative exponent, while option C results from an incorrect calculation of the exponents. Exam tip: When a power is raised to another power, multiply the exponents.

Using the power-of-a-power rule \((a^m)^n=a^{mn}\), \((2^{-2})^2=2^{(-2)\times2}=2^{-4}=\frac{1}{16}\). Therefore, option A is correct. Option B is incorrect because it changes the negative exponent to a positive one. Exam tip: when a power is raised to another power, multiply the exponents.

For like bases in a quotient, subtract the exponent in the denominator from the exponent in the numerator: \(a^{3-1}b^{2-1}=a^2b\). Therefore, option A is correct. Option C is incorrect because it reduces both factors too far. Exam tip: use \(\frac{x^m}{x^n}=x^{m-n}\) for the same nonzero base; here \(a\neq0\) and \(b\neq0\) are assumed.

The power of a product follows the rule \((xy)^n=x^ny^n\). Therefore, \((ab)^2=a^2b^2\), so option A is correct. Options B and C apply the exponent to only one variable, while option D incorrectly changes multiplication into addition. Exam tip: when a product is raised to a power, apply that power to every factor.

Write 4 as \(2^2\). Then \(2^5 \times 4^2 = 2^5 \times (2^2)^2 = 2^5 \times 2^4 = 2^{5+4}=2^9\). Therefore, option A is correct. Option B results from adding the exponents incorrectly. Exam tip: when powers with the same base are multiplied, their exponents are added.

Rewrite 8 as a power of 2: 8 = 2^3, so 8^2 = (2^3)^2 = 2^6. Therefore, \(\frac{8^2}{2^4}=\frac{2^6}{2^4}=2^{6-4}=2^2=4\). Hence, option A is correct; option D results from considering only the denominator and ignoring the numerator. In exams, subtract the exponents when dividing powers with the same base.

Using the law \((ab)^n=a^n b^n\), we get \((x^2y^3)^2=(x^2)^2(y^3)^2=x^{2\times2}y^{3\times2}=x^4y^6\). Therefore, option A is correct. In option B, the exponent of \(x\) was not multiplied by 2, while option C leaves the exponent of \(y\) unchanged. Exam tip: when a product is raised to a power, apply that power to every factor and multiply the exponents.

When powers with the same base are multiplied, their exponents are added: \(3^{-1} \times 3^2 = 3^{-1+2} = 3^1 = 3\). Therefore, the correct answer is 3. Exam tip: do not add the bases; apply the law \(a^m \times a^n = a^{m+n}\).