Exponents

Use the power-of-a-power rule: \((a^{m})^{n}=a^{m\times n}\). So \((7^2)^2=7^{2\times2}=7^4=2401\). Option B (343) equals \(7^3\), option C (49) equals \(7^2\), and option D (2400) is numerically close but incorrect. Exam tip: For (a^m)^n multiply the exponents m and n before evaluating the base.

When powers with the same base are multiplied, their exponents are added: \(5^2 \times 5^3 = 5^{2+3} = 5^5 = 3125\). Therefore, 3125 is correct. The close distractor 625 equals \(5^4\), so it does not result from adding the exponents correctly. Exam tip: use \(a^m \times a^n = a^{m+n}\) only when the bases are the same.

When powers with the same base are divided, subtract the exponents: \(6^4 \div 6^2 = 6^{4-2}=6^2=36\). Therefore, 36 is correct. The value 6 is only \(6^1\), while 216 equals \(6^3\). Exam tip: remember \(a^m \div a^n=a^{m-n}\), where \(a\ne0\).

When powers with the same base are divided, subtract the exponents: \(8^5 \div 8^3=8^{5-3}=8^2=64\). Therefore, 64 is correct. 512 is the value of \(8^3\), not the result after subtracting the exponents. Exam tip: use \(a^m \div a^n=a^{m-n}\) only when the bases are the same and non-zero.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((3^3)^2=3^{3\times2}=3^6=729\). Option 243 equals \(3^5\), so it is not correct. Exam tip: when a power is raised to another power, multiply the exponents.

Since \(125=5^3\), the equation \(a^3=125\) gives \(a=5\). Option 4 is incorrect because \(4^3=64\), not 125. Exam tip: remember common perfect cubes such as \(3^3=27\), \(4^3=64\), and \(5^3=125\).

Evaluate the powers first: \(2^5=32\) and \(2^4=16\). Therefore, \(2^5+2^4=32+16=48\), so 48 is correct. The value 32 represents only \(2^5\); \(2^4\) still has to be added. Exam tip: In expressions with exponents, evaluate the powers before performing addition or subtraction.

Evaluate the powers first: \(3^4=81\) and \(3^2=9\). Therefore, \(3^4-3^2=81-9=72\). Option 81 is only the value of \(3^4\), not the result after subtraction. Exam tip: in expressions with mixed operations, evaluate exponents before addition or subtraction.

Since \(4 = 4^1\), subtract the exponents when dividing powers with the same base: \(4^3 \div 4^1 = 4^{3-1} = 4^2 = 16\). Therefore, 16 is correct. The value 64 is \(4^3\) before dividing by 4. Exam tip: use \(a^m \div a^n = a^{m-n}\) for powers with the same base.

When powers with the same base are multiplied, their exponents are added: \(5^2 \times 5^2 = 5^{2+2}=5^4=625\). Therefore, 625 is the correct answer. 25 is only the value of \(5^2\), while 3125 equals \(5^5\). Exam tip: for multiplication of the same base, add the exponents.

Any non-zero number raised to the power zero equals 1, so \(6^0=1\). Therefore, \(6^0+1=1+1=2\). Option 1 is only the value of \(6^0\); the given \(+1\) must also be added. Exam tip: remember that \(a^0=1\) for every non-zero base \(a\).

Since \(12^2=144\), the solutions of \(x^2=144\) are \(x=12\) and \(x=-12\). The question asks for the positive value, so the answer is \(12\). Since \(14^2=196\), 14 is not correct. Exam tip: For \(x^2=a\), the solutions are usually \(\pm\sqrt{a}\); for the positive value, take only \(\sqrt{a}\).

Evaluate the exponents first: \(7^2=49\) and \(5^2=25\). Therefore, \(7^2+5^2=49+25=74\). Option 64 may result from an incorrect evaluation of the squares. Exam tip: Find each square separately before adding.

First evaluate the powers: \(8^2=64\) and \(2^3=8\). Therefore, \(8^2-2^3=64-8=56\). The value 48 may result from incorrectly taking \(2^3\) as 16. Exam tip: In expressions with exponents, evaluate the powers before addition or subtraction.

Using the law of exponents, subtract the exponents when dividing powers with the same base: \(9^2 \div 9^2 = 9^{2-2} = 9^0 = 1\). Therefore, the correct answer is 1. Although 81 is the value of \(9^2\), here \(9^2\) is divided by itself. Exam tip: remember that \(a^0=1\) for every non-zero number \(a\).

Use the power-of-a-power rule: \((a^m)^n=a^{mn}\). Hence, \((2^2)^3=2^{2\times3}=2^6=64\). The value 32 may result from adding exponents incorrectly; in this rule, the exponents are multiplied. Exam tip: an exponent outside brackets applies to the entire power inside the brackets.

Here, \(10^3=1000\) and \(10^2=100\). Therefore, \(10^3-10^2=1000-100=900\). The option 800 can result from evaluating the powers incorrectly. Exam tip: evaluate each power separately before subtracting.

Given \(a^3=216\), taking the cube root of both sides gives \(a=\sqrt[3]{216}=6\), because \(6^3=6\times6\times6=216\). Option 5 is not correct because \(5^3=125\), not 216. Exam tip: To check a cube, multiply the number by itself three times.

First evaluate the powers: \(11^2=121\) and \(3^2=9\). Therefore, \(11^2+3^2=121+9=130\). Hence, option C is correct. 129 results from an addition error. Exam tip: evaluate each power first, then check the addition separately.

First evaluate the squares: \(12^2=144\) and \(4^2=16\). Therefore, \(12^2-4^2=144-16=128\), so 128 is correct. An answer such as 124 may result from an arithmetic error in subtraction. Exam tip: evaluate exponents before performing addition or subtraction.