Exponents

We need to find the cube of 34: \(34^3=34\times34\times34\). First, \(34\times34=1156\), and then \(1156\times34=39304\). Hence, the correct answer is 39304. A value such as 38304 can result from an arithmetic error in multiplication. Exam tip: to find a cube, first square the number and then multiply by the same number.

The square of a number is obtained by multiplying it by itself. Thus, \(53^2=53\times53=2809\). Therefore, 2809 is correct. Although 2709 may look close, it is not obtained on multiplication. Exam tip: use \((50+3)^2=50^2+2\times50\times3+3^2\), which gives \(2500+300+9=2809\).

The cube means multiplying the number three times. First compute the square: \(35^2=35\times35=1225\). Then \(35^3=1225\times35=(1225\times30)+(1225\times5)=36750+6125=42875\). Option C (42775) is only 100 less — a typical arithmetic slip when adding partial products. Thus the correct value is 42875. Exam tip: compute the square first and then multiply by the base number to reduce mistakes in multi-digit multiplication.

Here, 54^2 means the square of 54, i.e. 54 × 54. Since 54 × 54 = 2916, option D is correct. A value such as 2906 may result from an error in multiplication, but the correct product is 2916. Exam tip: While finding a square, add the partial products carefully.

An exponent of 3 means multiplying 36 three times: \(36^3=36\times36\times36\). First, \(36\times36=1296\), and then \(1296\times36=46656\). Therefore, the correct answer is 46656. A value such as 45656 can result from an error in the final multiplication. Exam tip: For a cube, find the square first and then multiply once more by the base.

The square of a number is obtained by multiplying the number by itself. Thus, \(55^2=55\times55=3025\), so 3025 is correct. 2925 is not the correct product for the square of 55. Exam tip: You can verify it using \((50+5)^2=50^2+2\times50\times5+5^2=2500+500+25=3025\).

When powers with the same base are multiplied, their exponents are added: \(7^1\times7^2=7^{1+2}=7^3=343\). Therefore, 343 is correct. \(49\) is only the value of \(7^2\), so it is not correct. Exam tip: for multiplication with the same base, add the exponents, not the bases.

\(56^2\) means \(56\times56\). Using \((50+6)^2=50^2+2\times50\times6+6^2=2500+600+36=3136\), the correct value is 3136. An option such as 3126 results from an error in multiplication or addition. Exam tip: Use the identity \((a+b)^2\) to calculate squares quickly.

We need the cube of 37: \(37^3=37\times37\times37\). First, \(37\times37=1369\), and then \(1369\times37=50653\). Therefore, 50653 is correct. The other values can result from errors in multiplication. Exam tip: to find a cube, first square the number and then multiply by the same number.

The square of a number is obtained by multiplying it by itself. \(57^2=57\times57=57\times(50+7)=2850+399=3249\). Therefore, 3249 is the correct answer. A value such as 3149 can result from an error in multiplication, since \(57\times7=399\). Exam tip: You can also calculate \(57^2\) quickly using \((a+b)^2=a^2+2ab+b^2\).

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 \(65^2=(60+5)^2=3600+600+25=4225\).