Exponents

The governing concept is converting a set-builder description into roster form. The strict inequalities require prime numbers greater than 10 and less than 30. Testing the primes in that interval gives 11, 13, 17, 19, 23, and 29. Number 21 is composite, 31 is outside the interval, and 10 and 30 are excluded by strict inequalities. Therefore option A is correct.

The decimal expansion is infinite and, according to the question, does not settle into any repeating block. A rational number must have a terminating decimal or an eventually repeating decimal, so this number is not rational. It is therefore an irrational real number. Option B is correct. An integer is rational, while a terminating decimal ends after finitely many places and cannot describe this expansion.

The key classification rule is that rational numbers have decimal expansions which terminate or eventually repeat. This expansion continues without end, and the question explicitly says that no fixed repeating block exists. It therefore represents an irrational real number, making option B correct. It cannot be an integer because every integer is rational, and it is not terminating because digits continue indefinitely.

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.

Here, 2 is the base and 3 is the exponent. Therefore, \(2^3=2\times2\times2=8\), so 8 is correct. A common mistake is choosing 6 by calculating \(2\times3\), but an exponent does not mean multiplication by the exponent. Exam tip: In \(a^n\), multiply the base \(a\) by itself \(n\) times.

The exponent 2 means that 5 is multiplied by itself: \(5^2=5\times5=25\). Option 10 is the product of 5 and 2, but it is not the value of \(5^2\). Exam tip: \(a^2\) always means \(a\times a\).

An exponent of 2 means that 10 is multiplied by itself twice: \(10^2=10\times10=100\). Therefore, 100 is correct. The number 20 is twice 10, but it is not the value of \(10^2\). Exam tip: in \(a^n\), n tells how many times the base a is multiplied by itself.

The exponent 2 means that 3 is multiplied by itself twice: \(3^2=3\times3=9\). Therefore, 9 is correct. The value 6 comes from multiplying 3 by 2, but an exponent represents repeated multiplication. Exam tip: Check \(a^2\) by writing it as \(a\times a\).

An exponent of 2 means multiplying 4 by itself: \(4^2=4\times4=16\). Therefore, 16 is correct. The number 8 is twice 4, not its square. Exam tip: Check \(a^2\) by writing it as \(a\times a\).

By the exponent rule, any number raised to the power 1 remains the same number. Therefore, \(7^1=7\). Option 1 is the value of \(7^0\), while 49 is the value of \(7^2\). Exam tip: Remember \(a^1=a\) and \(a^0=1\), where \(a\ne0\), as different rules.

For any positive integer exponent, multiplying 1 by itself any number of times still gives 1. Therefore, \(1^8=1\). The number 8 is the exponent, not the value. Exam tip: \(1^n=1\) for every positive integer \(n\).

An exponent of 4 means that 2 is multiplied by itself four times: \(2^4=2\times2\times2\times2=16\). Therefore, 16 is the correct answer. The value 8 is \(2^3\), so it is a close but incorrect option. Exam tip: in \(a^n\), multiply the base \(a\) by itself \(n\) times.

An exponent of 2 means that 9 is multiplied by itself: \(9^2=9\times9=81\). Therefore, 81 is the correct answer. 90 is not the square of 9. Exam tip: Check \(a^2\) by writing it as \(a\times a\).

When powers with the same base are multiplied, their exponents are added: \(2^2\times2^3=2^{2+3}=2^5=32\). Therefore, 32 is correct. Getting 16 results from incorrectly multiplying the exponents. Exam tip: add exponents when multiplying like bases, and subtract them when dividing.

The exponent 3 means that the base 3 is multiplied by itself three times: \(3^3=3\times3\times3=27\). Therefore, 27 is correct. 9 is the value of \(3^2\), while 81 is the value of \(3^4\). Exam tip: in \(a^n\), n tells how many times a is used as a factor.

An exponent of 2 means that 6 is multiplied by itself: \(6^2=6\times6=36\). Therefore, 36 is correct. The value 12 is twice 6, not its square. Exam tip: Rewrite \(a^2\) as \(a\times a\) before calculating.

An exponent of 2 means that 8 is multiplied by itself: \(8^2=8\times8=64\). Therefore, the correct answer is 64. Note that 16 is \(4^2\), not the square of 8. Exam tip: Rewrite \(a^2\) as \(a\times a\) before calculating.

An exponent of 3 means that 10 is multiplied by itself three times: \(10^3=10\times10\times10=1000\). Therefore, 1000 is correct. The value 100 is a close distractor because it equals \(10^2\), not \(10^3\). Exam tip: In \(a^n\), n tells how many times a is used as a factor.