Exponents

Since 8=2^3, we get 8^x=(2^3)^x=2^{3x}. Thus 2^{3x}=2^{12}, so the exponents are equal: 3x=12, giving x=4. Exam tip: rewrite powers with a common base before comparing their exponents.

For powers with the same base, exponents are added during multiplication and subtracted during division. Thus, \(\frac{7^4\times7^{-2}}{7}=7^{4+(-2)-1}=7^1=7\). Therefore, option B is correct. Exam tip: write the denominator 7 as \(7^1\) before applying the laws of exponents.

Using the rule for fractional exponents, \(64^{2/3}=(\sqrt[3]{64})^2\). Since \(\sqrt[3]{64}=4\), we get \(4^2=16\). Therefore, option B is correct. Option A is only the cube root and does not include the required squaring. Exam tip: For \(a^{m/n}\), it is often easiest to take the \(n\)th root first and then raise the result to the \(m\)th power.

Writing 125 as a power of 5 gives \(125=5^3\). Therefore, \(5^{2x}=5^3\), and since the bases are equal, their exponents must be equal: \(2x=3\). Hence, \(x=\frac{3}{2}\). Exam tip: When powers have the same base, equate their exponents directly.

Factor out \(3^4\) from the numerator: \(3^5-3^4=3^4(3-1)=2\cdot3^4\). Therefore, \(\frac{3^5-3^4}{3^4}=\frac{2\cdot3^4}{3^4}=2\), so option B is correct. Exam tip: rewrite \(3^5\) as \(3\cdot3^4\) before cancelling; cancelling terms across a subtraction would incorrectly give option 1.

Express both numbers as powers of 3: \(27=3^3\) and \(9=3^2\). Thus, \(27^x=9\) becomes \((3^3)^x=3^2\), so \(3^{3x}=3^2\). Since the bases are equal, their exponents must be equal: \(3x=2\), giving \(x=\frac{2}{3}\). Therefore, option A is correct. Option B, \(\frac{3}{2}\), results from reversing the comparison of the exponents. Exam tip: Rewrite both sides with the same base before comparing exponents.

Since \(4=2^2\), we have \(4^5=(2^2)^5=2^{10}\). Therefore, \(\frac{4^5}{2^8}=\frac{2^{10}}{2^8}=2^{10-8}=2^2=4\). Hence, option B is correct. Exam tip: when dividing powers with the same base, subtract the exponents; choosing 8 results from applying this rule incorrectly.

Here, (9^{1/2}=\sqrt{9}=3) and (16^{1/2}=\sqrt{16}=4). Therefore, ((3+4)^2=7^2=49). Hence, 49 is correct. Exam tip: convert an exponent of 1/2 into a square root first, simplify the terms, and then square their sum.

Since 16 = 2^4, the equation becomes 2^{x-2} = 2^4. Powers with the same base are equal only when their exponents are equal, so x - 2 = 4 and x = 6. Therefore, option C is correct. Exam tip: Rewrite the number on the right as a power with the same base before comparing exponents.

Using the laws of exponents for the common base 6, add the exponents in the numerator and subtract the exponent in the denominator: \(3+(-1)-2=0\). Therefore, \(6^0=1\), so option B is correct. Remember that the zero power of any non-zero number is 1.

Using the rule for fractional exponents, \(125^{2/3}=(\sqrt[3]{125})^2\). Since \(\sqrt[3]{125}=5\), the value is \(5^2=25\). Therefore, option C is correct. Exam tip: For \(a^{m/n}\), it is often easiest to take the \(n\)th root first and then raise the result to the \(m\)th power.

Express both numbers as powers of the same base: \(32=2^5\) and \(8=2^3\). Thus, \((2^5)^x=2^3\), giving \(2^{5x}=2^3\). Since the bases are equal, compare the exponents: \(5x=3\), so \(x=\frac{3}{5}\). Therefore, option B is correct. Exam tip: In exponential equations, rewrite both sides with the same base and then equate their exponents.

Since \(9=3^2\), we have \(9^{-1}=3^{-2}\). Using the division law for powers with the same base, \(a^m\div a^n=a^{m-n}\), we get \(3^{-2}\div3^{-4}=3^{-2-(-4)}=3^2=9\). Therefore, option A is correct. In exams, take special care that subtracting \(-4\) becomes addition of 4.

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.