Exponents

Compute the powers first: \(28^2 = 784\) and \(17^3 = 4913\). Adding them gives \(784 + 4913 = 5697\). The closest distractor 5687 would result from mistakenly taking \(17^3=4903\) (10 less), while 5707 could come from adding an extra 10. Exam tip: evaluate each power separately and then add; use the units digit check (4+3=7) to quickly verify the last digit of the sum.

Evaluate the powers first: 16^3 = 4096 and 16^2 = 256. Therefore, 16^3 - 16^2 = 4096 - 256 = 3840, so option A is correct. Exam tip: calculate both powers before subtracting; 4096 is only the value of 16^3, not of the complete expression.

We are given \(h^3=15625\). Since \(25^3=25\times25\times25=15625\), it follows that \(h=25\). The cubes of the other options are not equal to 15625. Exam tip: When the cube of a number is given, find its cube root and verify the result by cubing it.

For division of powers with the same base, subtract the exponents: \(18^3 \div 18^2 = 18^{3-2} = 18^1 = 18\). Therefore, option A is correct. The answer 324 results from applying an incorrect operation instead of subtracting the exponents. Exam tip: add exponents when multiplying like bases and subtract them when dividing like bases.

Evaluate the powers first: 29^2 = 841 and 18^3 = 18 × 18 × 18 = 5832. Therefore, 841 + 5832 = 6673, so option B is correct. In such questions, calculate each power separately before performing the addition.

Use the power-of-a-power rule: \((a^m)^n = a^{mn}\). Therefore \((7^2)^3 = 7^{2\times3} = 7^6 = 117649\). Option A (16807) equals \(7^5\), B (823543) equals \(7^7\), and D (2401) equals \(7^4\), so they are incorrect. Exam tip: multiply the exponents first for expressions of the form \((a^m)^n\).

When powers with the same base are divided, their exponents are subtracted: \(a^m \div a^n=a^{m-n}\). Therefore, \(3^7 \div 3^3=3^{7-3}=3^4=81\). Hence, option C is correct. Exam tip: subtract exponents in division, whereas exponents are added when multiplying powers with the same base; therefore, 27 and 243 are not correct.

Given \(x=2^3=8\). Therefore, \(x^2=8^2=64\), so option B is correct. Note that \(x^2\) means squaring the complete value of \(x\), not squaring only the base 2 in \(2^3\). Exam tip: Using the exponent rule, \((2^3)^2=2^{3\times2}=2^6=64\) gives the answer directly.

When powers with the same non-zero base are divided, their exponents are subtracted: \(10^4 \div 10^2 = 10^{4-2}=10^2=100\). Therefore, the correct answer is 100. Exam tip: subtract exponents when dividing like bases; do not add them.

When powers with the same base are multiplied, their exponents are added: \(4^3 \times 4^2=4^{3+2}=4^5\). Since \(4^5=1024\), the correct answer is 1024. Exam tip: for \(a^m \times a^n\), use \(a^{m+n}\); do not multiply the exponents.

The zeroth power of any non-zero number is 1, so 7^0=1. Also, 7^1=7. Therefore, (7^0+7^1)=1+7=8, making option C correct. Exam tip: evaluate each power first and then perform the indicated addition; do not add the exponents directly.

Using the quotient rule for equal exponents, \(9^2 \div 3^2 = (9 \div 3)^2 = 3^2 = 9\). Therefore, the correct answer is 9. Exam tip: \(a^n \div b^n = (a \div b)^n\) can be applied when the exponents are equal.

For powers with the same base, exponents are added during multiplication and subtracted during division. Therefore, \((2^4 \times 2^3)\div 2^5 = 2^{4+3-5}=2^2=4\). Hence, option B is correct. Exam tip: Apply the laws of exponents before evaluating the final power.

By the zero-exponent rule, \(a^0=1\) for every non-zero number, so \(6^0=1\). Therefore, \(6^2 \times 6^0=36 \times 1=36\). Option B is incorrect because it does not evaluate \(6^2\). Exam tip: the zero power of any non-zero number is always 1.

Evaluate the powers first: \(8^2=64\) and \(2^5=32\). Therefore, \(8^2-2^5=64-32=32\), so option C is correct. Remember that exponentiation is performed before subtraction.

First evaluate the powers: \(3^4=81\) and \(3^2=9\). Therefore, \(3^4+3^2=81+9=90\), so option B is correct. Remember that powers cannot be added directly in an addition expression; evaluate each power separately and then add the results.

Since \(2^6=64\) and \(8=2^3\), we get \(8^2=(2^3)^2=2^6=64\). Therefore, option B is correct. Option A gives \(4^2=2^4\), while options C and D are much larger. In the exam, use the exponent rule \((a^m)^n=a^{mn}\).

When powers with the same base are divided, their exponents are subtracted: \(5^3 \div 5^1=5^{3-1}=5^2=25\). Therefore, the correct answer is 25. Remember to write the divisor 5 as \(5^1\) before applying the exponent rule.

Using the quotient rule for equal exponents, \\(a^n \div b^n = (a \div b)^n\\). Thus, \\(12^2 \div 6^2 = (12 \div 6)^2 = 2^2 = 4\\). Therefore, the correct answer is 4. Exam tip: do not divide only by 6; both the base and its square must be handled according to the exponent rule.

The first power of a non-zero number is the number itself, so \(11^1=11\). Also, the zero power of any non-zero number is 1, so \(11^0=1\). Therefore, \(11^1+11^0=11+1=12\). Exam tip: remember that \(a^0=1\) for \(a\neq0\).