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The order of operations requires exponents first, then multiplication and division from left to right, then addition and subtraction from left to right. Here 2^2 = 4, so 4 × 4 = 16 and 6 ÷ 3 = 2; thus 3 + 16 - 2 = 17.
A prime number has exactly two positive factors, 1 and itself. The number 59 is not divisible by 2, 3, 5, or 7, so it is prime, while 51 = 3 × 17, 57 = 3 × 19, and 63 = 3 × 21.
Let x = 0.363636... . Then 100x = 36.363636... , so 99x = 36 and x = 36/99 = 4/11. A two-digit repeating block can be placed over 99 and then simplified.
Starting at -180 feet, a descent of 75 feet gives -180 - 75, and a rise of 120 feet gives -180 - 75 + 120. This equals -135, which represents 135 feet below sea level.
An even divisor must contain at least one factor of 2. The exponent of 2 can be 1, 2, 3, or 4 (four choices); the exponent of 3 can be 0, 1, or 2 (three choices); and the exponent of 5 can be 0 or 1 (two choices). Therefore, 4 × 3 × 2 = 24 even positive divisors exist.
In 58,304, the digit 8 is in the thousands place, so its place value is 8 × 1,000 = 8,000. The digit’s face value is 8, but the question asks for its value in the numeral.
The additive inverse of a number is the number that adds to it to make zero. Since -6 + 6 = 0, the additive inverse of -6 is 6.
To find two-thirds of 3/4 cup, multiply the fractions: (3/4) × (2/3) = 6/12 = 1/2. So the smaller batch uses 1/2 cup of flour.
Dividing 0.6 by 0.15 is equivalent to dividing 60 by 15 after multiplying both numbers by 100. The quotient is 4, which can be checked by 0.15 × 4 = 0.6.
Write n = 7a + 3. The condition n ≡ 2 (mod 5) gives a ≡ 2 (mod 5), so n = 35b + 17. The condition n ≡ 4 (mod 9) then gives b ≡ 4 (mod 9), and the least positive solution is 35(4) + 17 = 157. Indeed, 157 leaves remainders 2, 3, and 4 when divided by 5, 7, and 9.
A number rounds to 4,700 when it lies from 4,650 through 4,749 inclusive, because 4,650 is exactly halfway and rounds up. Among the choices, only 4,652 lies in this interval. The others fall outside the interval and round to 4,600 or 4,800.
A number is divisible by 3 when its digit sum is a multiple of 3, and by 9 when its digit sum is a multiple of 9. The digit sum of 4,125 is 12, which is a multiple of 3 but not of 9. Therefore 4,125 is divisible by 3 but not by 9.
Using a common denominator of 24 gives 5/8 = 15/24, 2/3 = 16/24, and 7/12 = 14/24. Comparing the numerators 14, 15, and 16 gives the order 7/12, 5/8, 2/3. Thus the first list is the only correct ordering.
In base five, the place values from left to right are 5^2 = 25, 5^1 = 5, and 5^0 = 1. Therefore 324_5 = 3×25 + 2×5 + 4×1 = 75 + 10 + 4 = 89. The base-ten value is 89.
The runners are together at the starting point after a time that is a common multiple of 84 and 120 seconds. Factorizing gives 84 = 2^2 × 3 × 7 and 120 = 2^3 × 3 × 5, so the least common multiple is 2^3 × 3 × 5 × 7 = 840. Therefore they next meet at the starting point after 840 seconds.
An irrational number cannot be written as a ratio of two integers. Here √16 equals 4, 0.25 equals 1/4, and -3/5 is already a ratio, so all three are rational. Since 7 is not a perfect square, √7 is irrational.
Three moves of 4 units left represent adding -4 three times: (-4) + (-4) + (-4) = -12. Equivalently, 3 × (-4) = -12, so the final position is -12.
The number of pieces is the total length divided by the length of one piece: (7/8) ÷ (1/16). Dividing by 1/16 is the same as multiplying by 16, so (7/8) × 16 = 14.
The greatest number of identical sets must divide both 84 and 126, so it is their greatest common divisor. Since 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7, the GCD is 2 × 3 × 7 = 42.
There are floor(999/6) = 166 positive multiples of 6 below 1,000. Multiples of both 6 and 8 are multiples of lcm(6, 8) = 24, and there are floor(999/24) = 41 of them. Subtracting gives 166 - 41 = 125.
The multiplicative inverse of a nonzero number a is the number b such that a × b = 1. For -2/3, the reciprocal is -3/2 because (-2/3) × (-3/2) = 6/6 = 1. The negative sign must remain, since the original number is negative.
Multiplying by 10^3 moves the decimal point three places to the right. Starting with 5.04, the decimal point moves past 0 and 4 and then one more place, giving 5,040. Equivalently, 5.04 × 1,000 = 5,040.
The distributive property states that a(b + c) = ab + ac, and it can be applied in reverse to factor a common multiplier. Here both products share the factor 7, so 7 × 34 + 7 × 16 = 7(34 + 16). The expression then equals 7 × 50 = 350.
The fraction -3/4 equals -0.75. On a number line, numbers to the right are greater, and for negatives a number closer to zero is greater. Of the choices, only -0.70 lies to the right of -0.75.
Because the integer is divisible by 4 but not by 8, its prime factorization has exactly 2^2. If the odd part has d positive divisors, the total number of divisors is (2+1)d = 12, so d = 4. The smallest odd number with exactly four divisors is 3 × 5 = 15, giving 2^2 × 15 = 60; the other choices are larger or violate the condition about 8.
The prime factorization of 84 uses only prime factors. Since 84 = 4 × 21 = 2² × 3 × 7, the correct expression is 2² · 3 · 7. The other choices each contain a composite factor such as 14, 4, or 6.
Percent means per hundred, so 0.75% equals 0.75/100. Dividing by 100 moves the decimal point two places to the left, which gives 0.0075. Thus 0.75% is equivalent to 0.0075.
First simplify the parentheses to get 24 ÷ 8 × 2 − 16. Division and multiplication are done left to right, giving 3 × 2 − 16 = 6 − 16 = −10. The exponent is applied before the subtraction.
Let x = 0.272727... . Then 100x = 27.272727... . Subtracting x from 100x gives 99x = 27, so x = 27/99 = 3/11. This confirms that 0.272727... equals 3/11.
Write each divisor as 2^a 3^b 5^c with 0 ≤ a ≤ 3, 0 ≤ b ≤ 2, and 0 ≤ c ≤ 2. A divisor is a perfect square exactly when all three exponents are even. That gives 2 choices for a (0 or 2), 2 choices for b (0 or 2), and 2 choices for c (0 or 2), for 2 × 2 × 2 = 8 divisors.
In a base-ten numeral, each digit's value depends on its place. The 2 in 6,208 is in the hundreds place, so its value is 2 × 100 = 200.
An increase of 12°F means adding 12 to the starting temperature: −7 + 12 = 5. The result is 5°F, which is 12 degrees above −7°F on the number line.
The number of muffins increases by a factor of 30/12 = 5/2. Multiplying the milk amount by that factor gives (5/2) × (3/4) = 15/8 = 1 7/8 cups.
Dividing by 0.04 is equivalent to multiplying by 100/4, or 25. Therefore 3.6 ÷ 0.04 = 3.6 × 25 = 90.
The units digits of powers of 7 repeat in the cycle 7, 9, 3, 1, with period 4. Since 2024 is divisible by 4, 7^2024 has the same units digit as 7^4, which is 1.
The absolute value of a number is its distance from zero, so |-9| equals 9. The other expressions have values -12, 0, and -|-4| = -4, all of which are less than 9. Therefore, |-9| has the greatest value.
The term 3 × 10,000 contributes 30,000, the term 7 × 100 contributes 700, and the term 5 × 1 contributes 5. Combining these gives 30,000 + 700 + 5 = 30,705. The thousands and tens places are zero because no terms represent those place values.
The next simultaneous start occurs after the least common multiple of 18 and 24. The prime factorizations are 18 = 2 × 3² and 24 = 2³ × 3, so the LCM is 2³ × 3² = 72. Therefore, the shifts next begin together 72 minutes later.
Multiply numerators and denominators: (-3 × 10)/(5 × 9) = -30/45. Dividing numerator and denominator by 15 simplifies the product to -2/3. The product is negative because one factor is negative and the other is positive.
Each trailing zero comes from a factor of 10, which requires one factor of 2 and one factor of 5. In 30!, factors of 2 are more plentiful than factors of 5, so count the factors of 5: floor(30/5) + floor(30/25) = 6 + 1 = 7. Thus, 30! has 7 trailing zeros.
The absolute value of a number is its distance from zero, so |−4.6| = 4.6 and |2.9| = 2.9. Adding these nonnegative distances gives 4.6 + 2.9 = 7.5.
The decimal 0.45 means 45 hundredths, or 45/100. Dividing numerator and denominator by the greatest common factor, 5, gives 9/20, which is in simplest form.
A negative number squared is positive, so (−3)^2 = 9. A negative number cubed is negative, so (−2)^3 = −8. Subtracting −8 is the same as adding 8, and 9 + 8 = 17.
Divide the coefficients: 2.4 ÷ 6 = 0.4, and subtract exponents: 10^5 ÷ 10^2 = 10^3. This gives 0.4 × 10^3, which is not in scientific notation because 0.4 is less than 1. Moving the decimal one place right and decreasing the exponent by one gives 4 × 10^2.
A perfect cube must have each prime exponent divisible by 3. The given factorization has exponents 4, 2, and 1, so the missing factors are 2^2, 3^1, and 5^2. Their product is 4 × 3 × 25 = 300.
Divisibility by 6 requires divisibility by both 2 and 3. The number 3,414 is even, and its digit sum is 3 + 4 + 1 + 4 = 12, which is divisible by 3, so it is divisible by 6.
A rational number can be written as a ratio of two integers, while an irrational number cannot. The square root of 20 is not a perfect square and cannot be simplified to an integer or a terminating or repeating decimal, so it is irrational.
The greatest common factor of 84 and 120 is 12. Factoring out 12 gives 84 + 120 = 12(7 + 10), and 7 and 10 have no common factor greater than 1.
Use a common denominator of 12: 3/4 = 9/12, 5/6 = 10/12, and 1/3 = 4/12. Then 9/12 + 10/12 − 4/12 = 15/12, which simplifies to 5/4.
Convert by expanding powers of 6 from left to right. The numeral 2152_6 means 2×6^3 + 1×6^2 + 5×6^1 + 2×6^0 = 432 + 36 + 30 + 2 = 500.