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In 4,782 the digit 7 sits in the hundreds place, so its value is 7 hundreds, which equals 700. The place values from right to left are ones, tens, hundreds, thousands.
Each digit is placed in its named position: 6 in the ten-thousands place, 0 in the thousands place, 3 in the hundreds place, 5 in the tens place, and 2 in the ones place. This gives 60,352. The zero must be written to hold the thousands place open.
Compare digits from left to right after the decimal point. All four numbers begin with 0.4, so look at the next place: 0.4 has nothing, 0.405 has 0, 0.45 has 5, and 0.5 has a larger whole tenth. The correct order is 0.4, 0.405, 0.45, 0.5.
Rewrite 3/4 with denominator 8 as 6/8, then add 6/8 + 1/8 = 7/8. Fractions must share a common denominator before their numerators can be added.
The factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24; the factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, and 36. The largest factor the two numbers share is 12.
Multiples of 6 are 6, 12, 18, 24, ... and multiples of 8 are 8, 16, 24, ... The smallest number appearing in both lists is 24.
Multiply 2 1/2 by 3: 2 x 3 = 6 and 1/2 x 3 = 3/2 = 1 1/2, so the total is 6 + 1 1/2 = 7 1/2 cups. Multiplying a mixed number by a whole number multiplies both parts.
The decimal 0.6 means 6 tenths, or 6/10, which simplifies to 3/5 by dividing numerator and denominator by 2. Equivalent fractions represent the same value.
Adding the place-value parts gives 4,000 + 300 = 4,300, then + 20 = 4,320, then + 5 = 4,325. Each addend fills one place in the four-digit number.
Multiplying by 100 moves the decimal point two places to the right, so 7.4 becomes 740. Each power of ten shifts the digits one place.
Divide 156 by 12: 12 x 13 = 156, so exactly 13 packs are needed. Since the division is exact, no rounding up is required.
When multiplying powers with the same base, add the exponents: 2^4 x 2^3 = 2^(4+3) = 2^7. The base stays the same; only the exponents combine.
An irrational number cannot be written as a fraction of two integers and has a non-repeating, non-terminating decimal. The square root of 2 is a classic irrational number, since no fraction squared equals exactly 2.
Follow order of operations: parentheses first gives 8 - 2 = 6, then multiplication and division left to right give 3 x 6 = 18 and 18 ÷ 6 = 3, and finally 5 + 3 = 8.
A 25% discount removes 0.25 x 40 = $10 from the original price, leaving 40 - 10 = $30. Alternatively, paying 75% of $40 gives $30 directly.
First -8 + 15 = 7. Subtracting a negative is the same as adding, so 7 - (-3) = 7 + 3 = 10.
Scientific notation writes a number as a value between 1 and 10 times a power of ten. Moving the decimal point in 0.00045 four places right gives 4.5, so the exponent is -4.
Since 6 = 2 x 3 and 2 and 3 share no common factor other than 1, a number is divisible by 6 exactly when it is divisible by both 2 and 3. Testing those two conditions is enough.
Dividing by a fraction is the same as multiplying by its reciprocal: 3/5 x 3/2 = 9/10. Multiply numerators together and denominators together.
Divide 84 by 2 to get 42, by 2 again to get 21, then by 3 to get 7. The prime factors are 2, 2, 3, and 7, giving 2^2 x 3 x 7.
The distance from -3.5 to each option is the absolute value of their difference. The difference between -3.5 and -3.1 is 0.4, the smallest of the four distances.
Square each fraction: (2/3)^2 = 4/9 and (3/2)^2 = 9/4. Their product is (4/9) x (9/4) = 1, since the numerators and denominators cancel completely.
Multiplying by 0.25 is the same as dividing by 4, so the equation is n/4 = 18. Multiplying both sides by 4 gives n = 72.
Simplify each expression: 2^5 ÷ 2^2 = 2^3 = 8, 2^3 x 2^0 = 8 x 1 = 8, (2^2)^2 = 2^4 = 16, and 2^6 ÷ 2^4 = 2^2 = 4. The greatest value is 16.
The ratio 3:5 has 8 parts in total. Each part is 40 ÷ 8 = 5 marbles, so blue marbles are 5 parts x 5 = 25.
The GCD is the largest number that divides both 24 and 36 without a remainder. The divisors of 24 include 1, 2, 3, 4, 6, 8, 12, 24; those of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36. The largest common divisor is 12.
The decimal 0.45 means 45 hundredths, which can be written as 45/100. Dividing numerator and denominator by 5 gives 9/20, which is the simplified form. Thus 9/20 is equivalent to 0.45.
To add fractions, find a common denominator. The least common multiple of 4 and 6 is 12. Rewrite 3/4 as 9/12 and 1/6 as 2/12. Adding gives 11/12, which cannot be simplified.
The additive inverse of a number is the value that, when added to it, gives zero. For -7, the additive inverse is 7 because -7 + 7 = 0. The other options are either the number itself or its multiplicative inverse.
Any nonzero number raised to the power of zero equals 1. This follows from the rule that a^m / a^m = a^(m-m) = a^0, and any nonzero number divided by itself is 1. Therefore 5^0 = 1.
The square root of a number is a value that, when multiplied by itself, gives the original number. Since 7 × 7 = 49, the principal square root of 49 is 7.
The distributive property states that a(b + c) = ab + ac. Applying this, 3(2x + 4) = 3·2x + 3·4 = 6x + 12.
To multiply fractions, multiply the numerators and multiply the denominators: (2×3)/(3×4) = 6/12. Simplifying 6/12 by dividing numerator and denominator by 6 gives 1/2.
Adding the hundredths: 0.75 + 0.25 = 1.00. The sum is exactly 1, which can be written as 1.00 to show two decimal places.
The multiplicative inverse (reciprocal) of a number is the value that, when multiplied by the number, gives 1. For 5, the reciprocal is 1/5 because 5 × 1/5 = 1.
When multiplying powers with the same base, add the exponents: a^m × a^n = a^(m+n). Here, 2^3 × 2^2 = 2^(3+2) = 2^5. The other options either multiply exponents or change the base incorrectly.
Dividing by a fraction is the same as multiplying by its reciprocal. So 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2.
To convert 3/8 to a decimal, divide 3 by 8. 8 goes into 3.000 three times (24), remainder 6; bring down 0 to make 60, 8 goes in 7 times (56), remainder 4; bring down 0 to make 40, 8 goes in 5 times (40), remainder 0. So 3/8 = 0.375.
Multiply 25 by 4 to get 100. Since 2.5 has one decimal place and 0.4 has one decimal place, the product has two decimal places: 1.00, which equals 1.0.
Following the order of operations, perform multiplication before addition and subtraction. First, 2 × 3 = 6. Then the expression becomes 5 - 6 + 4. Working left to right, 5 - 6 = -1, and -1 + 4 = 3.
First distribute the 2: 2(x + 3) = 2x + 6. Then subtract 4: 2x + 6 - 4 = 2x + 2.
Compute each power: (-3)^2 = 9 because a negative times a negative is positive. (-2)^3 = -8 because a negative cubed is negative. Then 9 + (-8) = 1.
Dividing 0.6 by 0.2 is the same as dividing 6 by 2 after multiplying both numbers by 10. 6 ÷ 2 = 3.
List the factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The common factors are 1, 2, 3, 6. The greatest is 6.
The least common denominator for 2, 3, and 6 is 6. Rewrite each fraction: 1/2 = 3/6, 1/3 = 2/6, and 1/6 remains 1/6. Adding gives 3/6 + 2/6 + 1/6 = 6/6 = 1.
A rational number can be expressed as a fraction of two integers. The repeating decimal 0.333... is equal to 1/3, so it is rational. The square roots of non-perfect squares (√2, √5) and π are irrational.
Compute each power: 2^2 = 4 and 3^2 = 9. Multiply: 4 × 9 = 36. Alternatively, use the rule a^n × b^n = (a×b)^n, so 2^2 × 3^2 = (2×3)^2 = 6^2 = 36.
Multiplying by 10^3 (which is 1000) moves the decimal point three places to the right. Starting with 1.2, moving the decimal three places gives 1200.
Multiply numerators and denominators: (3×10)/(5×9) = 30/45. Simplify by dividing numerator and denominator by 15: 30/45 = 2/3.
Solve for x: add 7 to both sides to get 3x = 18. Then divide both sides by 3 to get x = 6.