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In a base-ten numeral, each position has a value ten times that of the position to its right. The 7 in 3,472,915 occupies the ten-thousands place, so its value is 7 × 10,000 = 70,000.
Order of operations requires evaluating exponents before multiplication, and multiplication before addition. So 3² = 9, then 2 × 9 = 18, then 6 + 18 = 24.
Absolute value gives the distance of a number from zero, so |−9| = 9 and |4| = 4. Subtracting the absolute values gives 9 − 4 = 5.
A number is divisible by 6 exactly when it is divisible by both 2 and 3. The number 4,122 is even and its digits sum to 9, which is a multiple of 3, so 4,122 is divisible by 6.
Write each number in prime factored form: 24 = 2³ × 3 and 36 = 2² × 3². The GCF uses the smallest power of each shared prime, giving 2² × 3 = 12.
A fraction can be converted to a decimal by dividing the numerator by the denominator. Dividing 3 by 8 gives exactly 0.375, a terminating decimal.
Writing each decimal with the same number of places makes comparison direct: 0.580, 0.600, 0.605, and 0.560. Comparing hundredths and thousandths shows 0.560 is the smallest.
To round to the nearest tenth, look at the digit in the hundredths place. Since that digit is 7, which is 5 or greater, increase the tenths digit by one and drop the remaining digits, giving 4.7.
Write each number in prime factored form: 12 = 2² × 3 and 18 = 2 × 3². The LCM uses the greatest power of each prime that appears, giving 2² × 3² = 36.
Convert the percent to a decimal and multiply: 0.15 × 240 = 36. Equivalently, 10% of 240 is 24 and 5% is 12, and 24 + 12 = 36.
A unit rate is found by dividing the total cost by the total quantity. Here $8.75 ÷ 3.5 lb = $2.50 per pound, and checking gives 3.5 × $2.50 = $8.75.
A rise in temperature is modeled by adding a positive integer: −8 + 13. The sum has the sign of the number with the greater absolute value and the magnitude 13 − 8 = 5, so the result is 5 °C.
Set up the proportion 3/12 = x/30 and solve by cross-multiplying: 12x = 90, so x = 7.5. Equivalently, each cookie needs 1/4 cup, and 30 × 1/4 = 7.5.
Scientific notation requires a coefficient between 1 and 10 multiplied by a power of ten. Moving the decimal point in 0.00042 four places to the right gives 4.2, so the exponent is −4.
Percent change compares the amount of change to the original amount. The decrease is $20, and 20/80 = 0.25, which is a 25% decrease.
A prime factorization expresses the number as a product of primes only. Since 84 = 4 × 21 = 2² × 3 × 7, and 2, 3, and 7 are all prime, this is the correct form.
Converting to decimals makes the comparison clear: 3/5 = 0.6, 5/8 = 0.625, 7/12 ≈ 0.583, and 2/3 ≈ 0.667. The largest decimal is 0.667, so 2/3 is the greatest fraction.
An irrational number cannot be written as a ratio of two integers, and its decimal form neither terminates nor repeats. Since 2 is not a perfect square, √2 has a non-terminating, non-repeating decimal expansion and is irrational.
Distance on a number line is the absolute value of the difference between the two points. Computing |7 − (−4)| = |11| = 11 gives the distance.
When dividing powers with the same base, subtract the exponents: 5⁶ ÷ 5² = 5^(6−2) = 5⁴. Since 5⁴ = 625, the simplified value is 625.
Let x = 0.444…, then 10x = 4.444…. Subtracting gives 9x = 4, so x = 4/9, and dividing 4 by 9 confirms the repeating decimal 0.444….
Successive discounts multiply rather than add. After the first discount the price is 0.80 of the original, and after the second it is 0.75 × 0.80 = 0.60 of the original, which is a 40% discount.
First factor 360 = 2³ × 3² × 5. The number of divisors equals the product of one more than each exponent: (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24.
Dividing by a fraction multiplies by its reciprocal, so (3/4) ÷ (9/8) = (3/4)(8/9) = 24/36 = 2/3. Then (2/3)(2/3) = 4/9.
If the tens digit is t and the units digit is u, reversing the digits changes the value by 9(u − t). Setting 9(u − t) = 27 gives u − t = 3, and together with t + u = 11 this yields u = 7 and t = 4, so the number is 47; checking, 74 − 47 = 27.
Each position in a whole number is worth ten times the position to its right. The 7 sits in the hundred-thousands place, so its value is 7 × 100,000 = 700,000.
A prime number has exactly two factors, 1 and itself. The number 61 is not divisible by 2, 3, 5, or 7, so it has no factors other than 1 and 61.
The number of bouquets must divide both 48 and 60, so the greatest possible number is the greatest common factor of the two counts. Since 48 = 2⁴ × 3 and 60 = 2² × 3 × 5, the GCF is 2² × 3 = 12, giving 4 roses and 5 tulips per bouquet.
A 25% increase means the new value is 1.25 times the original, so write 1.25x = 60. Dividing both sides by 1.25 gives x = 48, and checking: 48 + 0.25(48) = 48 + 12 = 60.
A negative exponent indicates the reciprocal raised to the corresponding positive power, so (2/3)⁻² = (3/2)². Squaring gives 9/4.
A number that is both a perfect square and a perfect cube must be a sixth power of a whole number. Since 64 = 8² and 64 = 4³, it satisfies both conditions.
Write n = 7k + 5 for some whole number k. Then 3n = 21k + 15 = 7(3k + 2) + 1, so dividing 3n by 7 leaves a remainder of 1.
Substitute -2 for every x: 3(-2)^2 - 2(-2) + 1. Since (-2)^2 = 4 this becomes 12 + 4 + 1 = 17, because the term -2x becomes positive when x is negative.
The common difference is 4, so the nth term is a(n) = 7 + (n - 1)(4). Substituting n = 20 gives 7 + 19(4) = 7 + 76 = 83.
Slope is the change in y divided by the change in x: (5 - (-3)) / (6 - 2). That gives 8/4 = 2, so the line rises 2 units for every 1 unit it moves right.
Add 8 to both sides to get 5x = 35, then divide both sides by 5 to get x = 7. Checking: 5(7) - 8 = 27.
Each time the input increases by 1, the output increases by 3, so the rule has a rate of 3 and is of the form y = 3x + b. Using the pair (1, 5): 5 = 3(1) + b, so b = 2, giving y = 3x + 2.
"Three times a number n" is 3n, and "five less than" that quantity means subtract 5 from it. The order matters, so the expression is 3n - 5.
Distribute the 4 to get 8x - 12, then add the like term 5x. Combining 8x and 5x gives 13x, so the simplified expression is 13x - 12.
A geometric sequence has a constant ratio found by dividing any term by the previous term. Here 12 divided by 3 equals 4, and 48 divided by 12 also equals 4.
Rewrite the equation in slope-intercept form by isolating y: -2y = -4x + 10, so y = 2x - 5. The coefficient of x, 2, is the slope.
Use point-slope form: y - 5 = -3(x - 2). Distributing gives y - 5 = -3x + 6, so y = -3x + 11. Checking the point: -3(2) + 11 = 5.
Substitute 2x + 1 for y in the second equation: 3x + (2x + 1) = 16, so 5x = 15 and x = 3. Then y = 2(3) + 1 = 7.
Subtract 5 from both sides to get -2x > 6. Dividing both sides by -2 reverses the inequality symbol, so x < -3.
First apply the power to each factor: (2x^2)^3 = 8x^6. Then divide: 8/4 = 2 and x^6/x^4 = x^2, giving 2x^2.
Distribute each term: x(x - 7) + 4(x - 7) = x^2 - 7x + 4x - 28. Combining -7x and 4x gives x^2 - 3x - 28.
The first differences are 3, 5, 7, 9, and the second differences are constantly 2, so the pattern is quadratic. The next first difference is 11, and 26 + 11 = 37.
Direct variation has the form y = kx. Since 28 = k(7), the constant of variation is k = 4, so y = 4x and y = 4(12) = 48.
Substitute -2 for every x: (-2)^2 - 3(-2) + 2 = 4 + 6 + 2. The product -3 times -2 is positive 6, so the value is 12.
Apply the rule step by step: a(2) = 2(3) + 1 = 7, a(3) = 2(7) + 1 = 15, and a(4) = 2(15) + 1 = 31.