Exponents

For multiplication with the same base add the exponents. Thus \(6^1\times6^2=6^{1+2}=6^3=216\). Option A (36) is incorrect — it corresponds to the wrong operation of multiplying exponents: \(6^{1\times2}=6^2=36\). Exam tip: add exponents for multiplication of same base; subtract for division.

We need to find the cube of 30: \(30^3=30\times30\times30=900\times30=27000\). Therefore, 27000 is correct. The value 2700 has one zero too few. Exam tip: Write \(30=3\times10\), so \(30^3=3^3\times10^3=27\times1000\).

Here, \(48^2\) means \(48 \times 48\). Since \(48 \times 48 = 2304\), the correct answer is 2304. A value such as 2204 is not the product of 48 and 48. Exam tip: To find the square of a number, multiply it by itself.

An exponent of 3 means multiplying 31 by itself three times: \(31^3=31\times31\times31\). First, \(31\times31=961\), and then \(961\times31=29791\). Therefore, 29791 is correct. A value such as 30791 can result from a small multiplication error. In exams, find the square first and then multiply by the same number.

An exponent shows how many times a number is multiplied by itself. Thus, \(10^5=10\times10\times10\times10\times10=100000\), so 100000 is correct. The close distractor 10000 equals \(10^4\), not \(10^5\). Exam tip: \(10^n\) has 1 followed by \(n\) zeros.

Here, \(49^2\) means \(49\times49\). Using \((50-1)^2=50^2-2\times50\times1+1^2=2500-100+1=2401\), the correct answer is 2401. Although 2501 is close to \(50^2\), the square of 49 is 99 less than 2500. Exam tip: use \((a-b)^2=a^2-2ab+b^2\) to calculate squares near a convenient number such as 50.

An exponent of 3 means multiplying 32 by itself three times: \(32^3=32\times32\times32=1024\times32=32768\). Therefore, 32768 is correct. Note that 1024 is the value of \(32^2\), not \(32^3\). Exam tip: In \(a^n\), multiply the base \(a\) by itself \(n\) times.

To find the square of a number, multiply it by itself. Thus, \(51^2=51\times51=2601\), so 2601 is correct. A value such as 2501 does not result from multiplying 51 by 51. Exam tip: You can quickly verify it using \((50+1)^2=50^2+2\times50\times1+1^2\).

An exponent of 3 means that 33 is multiplied by itself three times: \(33^3=33\times33\times33\). First, \(33\times33=1089\), and then \(1089\times33=35937\). Therefore, 35937 is the correct answer. The other options do not match this product. Exam tip: To find a cube, first square the number and then multiply by the same number.

We need to find the square of 52: \(52^2=52\times52\). Thus, \(52\times52=52\times(50+2)=2600+104=2704\), so option D is correct. Option A, 2604, can result from a common addition error during multiplication. Exam tip: check a square using \((a+b)^2\) or the distributive property.

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 \((a+b)^2=a^2+2ab+b^2\).