Exponents

Compute stepwise: first square 38: \(38\times38=1444\). Then multiply by 38: \(1444\times38=1444\times(30+8)=43320+11552=54872\). So 54872 is correct. Option C (55872) is wrong because it differs by 1000 — a typical arithmetic slip (adding/subtracting a power of ten). Exam tip: for cubes, calculate the square first and then multiply by the base to reduce mistakes.

Here, \(58^2\) means \(58 \times 58\). Using \((60-2)^2 = 60^2 - 2\times60\times2 + 2^2 = 3600 - 240 + 4 = 3364\), we get 3364. Hence, the correct answer is 3364. An option such as 3354 can result from an arithmetic error in multiplication or subtraction. Exam tip: For numbers close to 60, use the identity \((a-b)^2\) to find squares quickly.

An exponent of 3 means multiplying 39 three times: \(39^3=39\times39\times39\). First, \(39^2=1521\). Then \(1521\times39=1521\times(40-1)=60840-1521=59319\). Hence, the correct answer is 59319. A value such as 58319 can result from an error in multiplication or subtraction. Exam tip: For numbers such as 39, use \(40-1\) to make the calculation quicker.

\(59^2\) means \(59 \times 59\). Using \((60-1)^2 = 60^2 - 2\times60\times1 + 1^2 = 3600 - 120 + 1 = 3481\), the correct value is 3481. A value such as 3581 results from not applying the subtraction term correctly. Exam tip: For numbers close to 60, use \((a-b)^2\) to find the square quickly.

Use the law of exponents: when bases are equal, add the exponents: \(a^{m}\times a^{n}=a^{m+n}\). Here base 10 with exponents 1 and 2 gives \(10^{1}\times10^{2}=10^{1+2}=10^{3}=1000\). Option B (100) would result from the mistake of multiplying or ignoring exponents (or taking only \(10^2\)). Options A and D are also incorrect and reflect common calculation errors. Exam tip: convert factors to powers of the same base and add exponents; a quick check: \(10\times100=1000\).

Here, \(40^3=40\times40\times40\). First, \(40\times40=1600\), and then \(1600\times40=64000\). Therefore, the correct answer is 64000. The value 1600 is only \(40^2\), so it is a close but incorrect option. Exam tip: An exponent of 3 means multiplying the number by itself three times.

\(61^2=61\times61=(60+1)^2=60^2+2\times60\times1+1^2=3600+120+1=3721\). Therefore, 3721 is correct. The option 3621 can result from missing the middle term \(120\). Exam tip: use \((a+b)^2=a^2+2ab+b^2\) to quickly find squares of numbers close to 60 or 100.

An exponent of 3 means multiplying 41 by itself three times: \(41^3=41\times41\times41\). First, \(41\times41=1681\), and then \(1681\times41=68921\). Therefore, 68921 is correct. A value such as 68811 can result from an addition or place-value error during multiplication. Exam tip: to find a cube, first square the number and then multiply by the same number.

\(62^2\) means \(62 \times 62\). Using \((60+2)^2\), we get \(60^2+2\times60\times2+2^2=3600+240+4=3844\). Therefore, 3844 is correct. The nearby distractor 3824 does not include the middle term \(240\) correctly. Exam tip: For squaring a two-digit number, use \((a+b)^2=a^2+2ab+b^2\).

An exponent of 3 means multiplying 42 by itself three times: \(42^3 = 42 \times 42 \times 42\). First, \(42 \times 42 = 1764\), and then \(1764 \times 42 = 74088\). Therefore, the correct answer is 74088. A value such as 73088 can result from an arithmetic error in multiplication. Exam tip: to find a cube, first find the square of the number and then multiply it by the same number.

63^2 means 63 multiplied by 63: 63 × 63 = 3969. Therefore, the correct answer is 3969. A number such as 3869 can result from an error in multiplication or addition. Exam tip: You can also verify it using (60 + 3)^2 = 3600 + 360 + 9 = 3969.

An exponent of 3 means multiplying 43 by itself three times: \(43^3=43\times43\times43\). First, \(43\times43=1849\), and then \(1849\times43=79507\). Therefore, the correct answer is 79507. A close option such as 79407 can result from an addition or place-value error during multiplication. Exam tip: to find a cube, first square the number and then multiply by the same number.

We need to find the square of 64: \(64^2=64\times64=4096\). Therefore, 4096 is the correct option. Although it is close, 3969 equals \(63^2\), not \(64^2\). Exam tip: To find a square, multiply the number by itself.

An exponent of 3 means that 44 is multiplied by itself three times: \(44^3=44\times44\times44\). First, \(44\times44=1936\), and then \(1936\times44=85184\). Therefore, 85184 is correct. A value such as 84184 can result from an error in multiplication or place value. Exam tip: to find a cube, first square the number and then multiply by the original number.

The square of a number is obtained by multiplying the number by itself. Thus, \(65^2=65\times65=4225\), so 4225 is the correct option. 4125 is too small, while 4325 and 4425 are too large. Exam tip: You can quickly check using \(65^2=(60+5)^2=3600+600+25=4225\).

An exponent of 3 means that 45 is multiplied by itself three times: \(45^3=45\times45\times45\). First, \(45\times45=2025\), and then \(2025\times45=91125\). Therefore, the correct answer is 91125. Note that 2025 is the value of \(45^2\), not \(45^3\). Exam tip: To find a cube, first find the square and then multiply it by the original number.

When powers with the same base are multiplied, their exponents are added: \(2^5 \times 2^3 = 2^{5+3} = 2^8 = 256\). Therefore, 256 is correct. 128 equals \(2^7\), so it does not result from adding the exponents correctly. Exam tip: add exponents when multiplying like bases; subtract them when dividing.

When powers with the same base are divided, subtract the exponents: \(5^6 \div 5^2 = 5^{6-2}=5^4=625\). Therefore, 625 is correct. Getting 25 would result from subtracting the exponents incorrectly. Exam tip: for division with the same base, subtract the denominator’s exponent from the numerator’s exponent.

Using the power-of-a-power rule, \((a^m)^n=a^{mn}\). Therefore, \((3^2)^3=3^{2\times3}=3^6=729\). Hence, 729 is the correct answer. Note that 243 equals \(3^5\), so it is not correct. Exam tip: multiply the exponents when a power is raised to another power.

When powers with the same base are divided, subtract the exponents: \(10^4 \div 10^4 = 10^{4-4} = 10^0 = 1\). Therefore, the correct answer is 1. It is not 0, because any non-zero number divided by itself equals 1. Exam tip: use \(a^m \div a^n = a^{m-n}\), where \(a \ne 0\).