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In the base-ten place-value system each position is ten times the value of the position to its right. In 4,728 the 7 sits in the hundreds place, so it represents 7 × 100 = 700.
The additive inverse of a number is the value that sums with it to give zero. Since −13 + 13 = 0, the additive inverse is 13, which is the opposite of −13 on the number line.
Multiply the amount per batch by the fraction of a batch: (3/4) × (2/3) = 6/12 = 1/2. So half a cup of oil is required.
Subtracting a negative is the same as adding its opposite, so 6 − (−4) = 10. Then adding −9 gives 10 − 9 = 1.
Comparing decimals requires aligning place values: 0.09 < 0.35 < 0.40 < 0.50. Writing 0.4 as 0.40 makes the tenths and hundredths comparison clear.
The prime factorizations are 36 = 2² × 3² and 60 = 2² × 3 × 5. Taking the lowest power of each shared prime gives 2² × 3 = 12.
25% equals 1/4, so 480 ÷ 4 = 120. Equivalently, 0.25 × 480 = 120.
When multiplying powers with the same base, add the exponents: 2⁵ × 2³ = 2^(5+3) = 2⁸. The base stays the same.
The discount is 0.20 × 60 = 12, so the sale price is 60 − 12 = 48. Alternatively, paying 80% of the price gives 0.80 × 60 = 48.
Absolute value gives distance from zero, so |−7| = 7. Also 3 − 9 = −6 and |−6| = 6, giving 7 + 6 = 13.
Average speed equals distance divided by time: 150 ÷ 2.5 = 60. Checking, 60 mph × 2.5 h = 150 miles.
An irrational number cannot be written as a ratio of integers. √2 has a non-terminating, non-repeating decimal expansion and is irrational, while the others are rational.
Dividing by a fraction means multiplying by its reciprocal: (5/6) × (3/2) = 15/12 = 5/4. The result can be left as an improper fraction.
If the original is x, then 1.15x = 92, so x = 92 ÷ 1.15 = 80. Checking, 80 × 1.15 = 92.
Scientific notation requires one nonzero digit before the decimal point. Moving the decimal four places right gives 4.2, so the exponent is −4.
List multiples: 8, 16, 24, … and 12, 24, 36, …; the first shared value is 24. Equivalently, LCM = (8 × 12) ÷ GCD = 96 ÷ 4 = 24.
Integers are closed under subtraction because the difference of any two integers is always an integer, including negatives. Division and non-integer rationals break the other claims.
First multiply: 2³ × 2⁻⁵ = 2⁻². Then divide by 2⁻¹ by subtracting exponents: 2^(−2 − (−1)) = 2⁻¹.
The sale price is 80% of the original, so 0.80x = 80 and x = 100. Checking, a 20% discount on $100 gives $80.
A fraction in lowest terms terminates exactly when its denominator has only 2 and 5 as prime factors. Since 8 = 2³, 1/8 = 0.125 terminates.
Order of operations: exponent first (2² = 4), then multiply and divide left to right, then add and subtract. This gives 3 + 16 − 2 = 17.
Subtracting a positive number from a negative number moves further left on the number line, so x − y is always negative; its sign is wrong. Rechecking: x − y with x negative and y positive gives a negative result, so the positive quantity is x × y? No - a negative times a positive is negative. The expression that must be positive is x − y only when |x| > y, which is not guaranteed. The correct guaranteed-positive expression is x × x, but among the options only y ÷ x is negative. The only option that is always positive is x + y? That can be negative. The guaranteed one is x − y? That is negative. Therefore the correct option is y ÷ x? No, that is negative. The option that must be positive is x × y? No. Among the listed options, x − y is negative, x + y may be negative, x × y is negative, and y ÷ x is negative. The intended answer is x − y being negative, so the question is flawed.
Numbers congruent to 3 mod 7 are 3, 10, 17, 24, 31, 38, …. Checking mod 5: 3→3, 10→0, 17→2. Wait, 17 gives remainder 2 mod 5 and remainder 3 mod 7, so 17 is the smallest. Rechecking 23: 23 mod 7 = 2, not 3. So the answer is 17.
Multiples of 3 below 100 are 3, 6, …, 99, giving 33 values. Those also divisible by 6 are the even multiples, 6, 12, …, 96, giving 16 values. Subtracting gives 33 − 16 = 17. Rechecking: 3, 9, 15, …, 99 form an arithmetic sequence with 17 terms. So the answer is 17.
Let the integers be n and n + 2 with n even. Then n(n + 2) = 288, so n² + 2n − 288 = 0, giving (n + 18)(n − 16) = 0. The positive solution is n = 16, and 16 × 18 = 288.
Since 12 = 4 × 3, any multiple of 12 is also a multiple of 4. Divisibility by 12 guarantees divisibility by every factor of 12, including 4, 3, 2, and 6.
A number with exactly two distinct positive factors is by definition prime. Composite numbers have at least three factors, and perfect squares have an odd number of factors (at least three when greater than 1).
Each minus sign applied to a negative reverses the sign. Starting with −5, the innermost −(−5) = 5, then the outer negative gives −5.
For the product to be divisible by 6 = 2 × 3, both a and b must be at least 1. The smallest such choice is a = 1, b = 1, giving 2 × 3 = 6.
(−3)² = 9 because a negative squared is positive. (−2)³ = −8 because a negative cubed is negative, so subtracting it gives 9 − (−8) = 17.
If n = 9k + 4, then 3n = 27k + 12 = 9(3k + 1) + 3. The remainder is 3, since 12 mod 9 = 3.
The reciprocal of −4 is −1/4. Its additive inverse is the number that sums with it to zero, namely 1/4.
For a factorization p^a × q^b × r^c, the number of positive factors is (a+1)(b+1)(c+1). Here that is (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24.
Multiplying by 0.25 and dividing by 0.5 equals multiplying by 0.25 ÷ 0.5 = 0.5. Dividing by one-half is the same as multiplying by 2, so 0.25 × 2 = 0.5.
The decimal 2/3 = 0.666…, which continues indefinitely. Since 0.666… exceeds 0.660, we have 2/3 > 0.66.
The pair 7 and 12 has product 84 and sum 19, so the difference is 12 − 7 = 5. Checking factor pairs of 84 confirms 7 and 12 is the only pair summing to 19.
If n ≡ 5 (mod 8), then n² ≡ 25 (mod 8). Since 25 = 3 × 8 + 1, the remainder is 1.
Multiples of 4 up to 200 number 50. Multiples of both 4 and 6 are multiples of 12, and there are 16 of these up to 200. So 50 − 16 = 34. Checking inclusion at the boundary: 200 is not a multiple of 12, so the count of 16 stands, giving 34.
A perfect square requires every prime exponent to be even. The exponent of 2 is already 4 (even), but the exponent of 3 is 3 (odd), so one more factor of 3 makes it 3⁴. The smallest multiplier is therefore 3.
The square of a prime p² has exactly three factors: 1, p, p². None of the squares of primes has six factors, so the condition is contradictory; among the choices, 25 = 5² fits the square-of-a-prime condition, while only 16 = 2⁴ and 36 = 2² × 3² have exactly five and nine factors respectively. The intended answer is the square of a prime, 25.
Substitute x = 5 into the rule: f(5) = 3(5) - 7 = 15 - 7 = 8. Evaluation of a linear function requires multiplying by the coefficient first, then applying the constant term.
A rational function is undefined wherever its denominator equals zero. Setting x - 4 = 0 gives x = 4, so every real number except 4 is allowed.
Compute (-3)² = 9 and 2(-3) = -6, then add: 9 - 6 = 3. Squaring a negative yields a positive value, which is the most common source of error.
Horizontal shifts act on the input and behave opposite to intuition: replacing x with x - 3 moves the graph right by 3. Vertical shifts, by contrast, are added outside the function.
x² - 9 is a difference of squares with a = x and b = 3, so it factors as (x - 3)(x + 3). Expanding confirms the middle terms cancel.
In slope-intercept form y = mx + b, the coefficient of x is the slope, so m = -2. A negative slope means the output falls as the input rises.
Composition applies g first: f(g(x)) = 2(x - 4) + 1 = 2x - 8 + 1 = 2x - 7. Order matters, since f ∘ g generally differs from g ∘ f.
Set y = 4x - 8, swap x and y to get x = 4y - 8, then solve for y: y = (x + 8)/4. The inverse undoes each operation in reverse order.
Complete the square: x² - 6x + 5 = (x - 3)² - 4, so the vertex is (3, -4). Equivalently, the vertex x-coordinate is -b/(2a) = 6/2 = 3.
The fraction is undefined when the denominator is zero: x² - 9 = 0 gives x = 3 or x = -3. Zeros of the numerator make the expression zero, not undefined.