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A negative power of ten shifts the decimal point to the left, and the exponent −3 moves it three places from 4.70. Moving the point three places left in 4.70 gives 0.00470, which equals 0.0047. The trailing zero in 4.70 does not change the place value.
Repeated division by the smallest primes gives 360 = 2 × 2 × 2 × 3 × 3 × 5. Collecting gives 2³ × 3² × 5, and every factor here is prime, as a prime factorisation requires.
Writing 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7, the HCF takes the lowest power of each shared prime: 2 × 3 × 7 = 42. Checking, 84 ÷ 42 = 2 and 126 ÷ 42 = 3, both whole numbers.
The next simultaneous return occurs after the lowest common multiple of 18 and 24. Since 18 = 2 × 3² and 24 = 2³ × 3, the LCM is 2³ × 3² = 72 minutes.
Brackets come first: 8 − 2 = 6. The index gives 6² = 36, then multiplication and division run left to right: 3 × 36 = 108 and 108 ÷ 4 = 27. Adding 5 gives 32.
After the first reduction the price is 80 × 0.85 = 68. The second reduction applies to 68, giving 68 × 0.90 = 61.20. Successive percentage changes multiply, so the overall reduction is 23.5%, not 25%.
Simple interest is calculated on the original principal only: I = 5000 × 0.04 × 3 = 600. Adding this to the principal gives 5000 + 600 = $5600.
An irrational number cannot be written as a fraction of two integers. The square root of 2 is non-recurring and non-terminating, so it is irrational, whereas the other three are rational.
The decimal 0.45 means 45 hundredths, so it equals 45/100. Dividing numerator and denominator by their highest common factor, 5, gives 9/20, which cannot be simplified further.
To divide by a fraction, multiply by its reciprocal: 3/4 × 5/2 = 15/8. The numerator and denominator share no common factor, so the fraction is already in its simplest form.
The flour share of 5 parts corresponds to 350 g, so one part is 350 ÷ 5 = 70 g. Sugar is 2 parts, giving 2 × 70 = 140 g.
The largest share is 5 parts and equals 45, so one part is 9. The total is 2 + 3 + 5 = 10 parts, giving 10 × 9 = 90.
A prime has exactly two factors, 1 and itself. Checking the candidates between 20 and 40, only 23, 29, 31 and 37 have no divisor other than 1 and themselves.
Writing 72 = 2³ × 3², the number of factors is found by adding one to each exponent and multiplying: (3 + 1)(2 + 1) = 12. Listing them confirms 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36 and 72.
In 3.472 the digits after the decimal point are tenths, hundredths and thousandths in order. The 7 sits in the second decimal place, so it represents 7 hundredths, that is 0.07.
The first significant figure is 4 (the leading zeros are not significant). The first three significant figures are 4, 7 and 3, and the next digit is 8, so the 3 rounds up to 4, giving 0.0474.
Divide the coefficients and subtract the indices: 4.8 ÷ 6.0 = 0.8 and 10⁵ ÷ 10³ = 10². Since 0.8 × 10² is not in standard form, adjust to 8 × 10¹.
Writing 12 = 2² × 3, 18 = 2 × 3² and 30 = 2 × 3 × 5, the LCM takes the highest power of each prime: 2² × 3² × 5 = 180. This is the smallest number divisible by all three.
Convert the time to hours: 2 hours 30 minutes is 2.5 hours. Average speed equals distance divided by time, so 150 ÷ 2.5 = 60 km/h.
The increase is 27 500 − 25 000 = 2500. Percentage increase equals increase divided by original, so 2500 ÷ 25 000 = 0.1, which is 10%.
When multiplying powers with the same base, add the indices: 2³ × 2⁵ = 2⁸. When dividing, subtract: 2⁸ ÷ 2⁴ = 2⁴, which equals 16.
Multiply the coefficients: 3 × 2 = 6. Add the indices: 4 + (−6) = −2. The result 6 × 10⁻² already has a coefficient between 1 and 10, so it is in standard form.
Compound growth multiplies the principal by 1.05 each year, so after two years the multiplier is 1.05² = 1.1025. Dividing 1102.50 by 1.1025 gives $1000.
A common multiple must be divisible by both 6 and 8, so it must be a multiple of their LCM, which is 24. Among the options, 48 = 24 × 2 is a multiple of 24 and hence of both 6 and 8.
Rounding 498 to one significant figure gives 500, and 21 gives 20. Their product is 500 × 20 = 10 000, which is a sensible estimate of the exact value 10 458.
Evaluate the prime factorisation directly: 2⁴ = 16, 3² = 9, so 16 × 9 = 144 and 144 × 5 = 720. The prime factor form gives the unique factorisation of the number, so no other value can match it.
√144 = 12 because 12 × 12 = 144, and ∛27 = 3 because 3 × 3 × 3 = 27. Subtracting gives 12 − 3 = 9.
Convert to decimals: 3/5 = 0.6 and 7/12 ≈ 0.5833, while the middle value is 0.58. Comparing 0.58 < 0.5833 < 0.6 gives the order 0.58, 3/5, 7/12.
A number has exactly three factors only when it is the square of a prime, because the factors are 1, p and p². The only such numbers below 20 are 4 and 9, and 4 is excluded by 1 < n, so n = 9.
Dividing 84 by primes gives 84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7, so a = 2, b = 1 and c = 1. Therefore a + b + c = 2 + 1 + 1 = 4.
A 25% discount leaves 75% of $64, which is $48. Adding 10% tax multiplies by 1.10, giving 48 × 1.10 = $52.80. The two percentage changes act on different amounts, so they do not cancel.
Compound interest multiplies the balance by 1.03 each year: 2000 × 1.03² = 2000 × 1.0609 = $2121.80. The second year's interest is earned on the first year's interest as well as the original deposit.
Moving the decimal point 24 places to the right gives 1.6, so the power of ten is 10⁻²⁴. In standard form a must satisfy 1 ≤ a < 10, so a = 1.6 and n = −24.
Each centimetre on the map represents 25 000 cm in reality, so 6 cm represents 150 000 cm. Dividing by 100 000 to convert centimetres to kilometres gives 1.5 km.
Rounding to 2 significant figures gives a half-unit interval of 0.0005 around 0.040, so the lower bound is 0.040 − 0.0005 = 0.0395. Any smaller value would round down to 0.039.
For the highest common factor, take the lower power of each shared prime: 2² × 3² = 4 × 9 = 36. The prime 5 and the prime 7 appear in only one number each, so they are not included.
The difference between B's and A's shares is 4 − 3 = 1 part, and this equals $28, so one part is $28. The total is 3 + 4 + 5 = 12 parts, giving 12 × 28 = $168.
Multiply the numerical parts and add the indices: 3.2 × 4.5 = 14.4 and 10² × 10¹ = 10³, giving 14.4 × 10³. Converting to standard form gives 1.44 × 10⁴.
Any common multiple of 4 and 6 must be a multiple of their lowest common multiple, which is 12. So n is always a multiple of 12, but not necessarily of 24.
For a number expressed as a product of prime powers, the number of positive factors is found by adding 1 to each exponent and multiplying these results. Here the exponents are 2, 1, and 3, so the number of factors is (2+1)(1+1)(3+1) = 3 × 2 × 4 = 24.
Subtract 3x from both sides to get 2x - 7 = 9, then add 7 to both sides to get 2x = 16. Dividing both sides by 2 gives x = 8. Substituting back confirms 33 = 33.
Multiply each term of the first bracket by each term of the second: 2x·x = 2x^2, 2x·5 = 10x, -3·x = -3x, -3·5 = -15. Combining 10x and -3x gives 7x, so the result is 2x^2 + 7x - 15.
We need two numbers whose product is +12 and whose sum is -7. The pair -3 and -4 satisfies both conditions, so the factorisation is (x - 3)(x - 4). Expanding confirms the middle term is -7x.
Gradient equals the change in y divided by the change in x: (11 - 3)/(6 - 2) = 8/4 = 2. The line rises 2 units for every 1 unit to the right.
Multiply the coefficients: 3 × 4 = 12. For the powers, add exponents of like bases: a^2·a = a^3 and b·b^3 = b^4. The simplified product is therefore 12a^3b^4.
Multiply both sides by 3 to obtain 3y = x - 4, then add 4 to both sides to isolate x. This gives x = 3y + 4.
Using the laws of indices, add exponents for multiplication and subtract for division: 2^(3+4-5) = 2^2. Since 2^2 = 4, the expression equals 4.
Subtract 4 from both sides to get -2x > 6. Dividing both sides by -2 reverses the inequality sign, giving x < -3.
For ax^2 + bx + c = 0, the sum of the roots equals -b/a. Here a = 1 and b = -5, so the sum is -(-5)/1 = 5. The roots are 2 and 3, whose sum is indeed 5.
Adding the two equations eliminates y and gives 3x = 12, so x = 4. Substituting into x - y = 1 gives 4 - y = 1, hence y = 3. Checking the first equation: 8 + 3 = 11.