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An irrational number cannot be written as a ratio of two integers. The square root of 50 lies between 7 and 8 and is not a perfect square, so it is irrational. The other choices all equal rational values.
Following order of operations, evaluate exponents first, then multiplication and division left to right, then addition and subtraction. This gives 25 − 12 + 4. Combining gives 17 − 2? No: 25 − 12 = 13, then 13 + 4 = 17? Recheck: 8 ÷ 2 = 4, so 25 − 12 + 4 = 17.
Multiply the coefficients: 2 × 3 = 6. Then add the exponents of the powers of ten: 3 + 4 = 7. The product is 6 × 10⁷, already in scientific notation.
Write each number in prime factors: 84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. The GCD uses the lowest power of each shared prime: 2 × 3 × 7 = 42.
For negative numbers, the value with the largest absolute value is smallest. Converting to decimals gives −0.9, −0.75, −0.667, and −0.5. Ordering from most negative to least negative yields −0.9, −3/4, −2/3, −0.5.
An increase of 25% means the new value equals 1.25 times the original. Divide 60 by 1.25 to find the original value, which is 48. Checking: 48 × 1.25 = 60.
When multiplying powers with the same base, add the exponents: x⁵ · x³ = x⁸. When dividing, subtract the exponents: x⁸ / x² = x⁶.
A fractional exponent a^(m/n) means the n-th root of a raised to the m-th power. The cube root of 27 is 3, and 3² = 9. So 27^(2/3) equals 9.
A negative exponent of −7 moves the decimal point seven places to the left. Starting from 4.5 and moving seven places gives 0.00000045.
Set up the proportion 5/2 = 35/x, where x is the grams of sugar. Cross-multiplying gives 5x = 70, so x = 14.
Scientific notation requires a coefficient between 1 and 10. Move the decimal five places right to get 4.56, so the exponent is −5. The result is 4.56 × 10⁻⁵.
Prime factorizations are 12 = 2² × 3 and 18 = 2 × 3². The LCM uses the highest power of each prime: 2² × 3² = 36.
If a rational plus an irrational number were rational, subtracting the rational part would make the irrational number rational, a contradiction. Therefore the sum must be irrational. For example, 2 + √3 is irrational.
Divide the coefficients: 3 ÷ 6 = 0.5. Subtract the exponents: 8 − 2 = 6, giving 0.5 × 10⁶. Converting to scientific notation gives 5 × 10⁵.
After a 20% markdown, the price is 0.8 of the original. After a further 25% markdown, the price is 0.75 × 0.8 = 0.6 of the original. A final price of 60% means a total discount of 40%.
Raise each factor in the product to the third power. The coefficient becomes 2³ = 8, and each exponent is multiplied by 3: x³·³ = x⁹ and y²·³ = y⁶. The result is 8x⁹y⁶.
The square root of 40 is between 6 and 7 because 6² = 36 and 7² = 49, and 40 is not a perfect square, so √40 is irrational and greater than 6.
Rewrite 81 as a power of 3: 81 = 3⁴. Then 3^(2x) = 3⁴, so 2x = 4 and x = 2.
If a prime factorization is p^a × q^b × r^c, the number of positive divisors is (a+1)(b+1)(c+1). Here that is (4+1)(2+1)(1+1) = 5 × 3 × 2 = 30.
Exponential growth by a constant percent uses the formula P(1 + r)^t. With r = 0.08 and t = 5, the expression is P(1.08)⁵.
Like terms combine by adding their coefficients while the variable part stays the same. The coefficients are 3 + 5 − 2 = 6, so the simplified result is 6x.
Add 7 to both sides to get 4x = 28, then divide both sides by 4 to isolate x. This gives x = 7, which checks because 4(7) − 7 = 21.
Slope 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 it moves right.
Using the distributive property, multiply each term of the first binomial by each term of the second: x² + 3x + 4x + 12. Combining the middle terms gives x² + 7x + 12.
Both 7 and −7 square to 49 because a negative number times a negative number is positive. Taking the square root of both sides therefore yields two solutions, x = 7 and x = −7.
Multiply the coefficients 2 · 5 = 10, then add the exponents of like bases: x³ · x⁴ = x^(3+4) = x⁷. The product is 10x⁷.
Substitute 5 for x in the rule: f(5) = 2(5) + 1 = 10 + 1 = 11. Function notation means the input value replaces every x in the expression.
Subtract 2 from both sides to get 3x < 12, then divide both sides by the positive number 3, which keeps the inequality direction. The result is x < 4.
This is a difference of two squares, since x² − 9 = x² − 3². The pattern a² − b² factors as (a − b)(a + b), giving (x − 3)(x + 3).
In slope-intercept form y = mx + b, the value b is the y-intercept, the point where x = 0. Here b = 6, so the line crosses the y-axis at (0, 6).
Adding the two equations eliminates y and gives 2x = 14, so x = 7. Substituting back into x + y = 10 gives y = 3, and both equations check.
The zeros occur where each factor equals zero. Setting x − 1 = 0 gives x = 1, and setting x − 5 = 0 gives x = 5, so the zeros are 1 and 5.
When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. Here 5 − 2 = 3, so the quotient is x³.
Rewrite the equation in slope-intercept form by solving for y: 5y = −2x + 10, so y = (−2/5)x + 2. Parallel lines share the same slope, so the required slope is −2/5.
Substitute −2 for x: (−2)² − 3(−2) = 4 + 6 = 10. The square of a negative number is positive, and subtracting a negative product adds.
Squaring a binomial means multiplying it by itself: (x + 2)(x + 2) = x² + 2x + 2x + 4. Combining the middle terms gives x² + 4x + 4.
Distribute to get 2x − 6 = 5x + 6, then subtract 2x from both sides to get −6 = 3x + 6. Subtracting 6 gives −12 = 3x, so x = −4.
Adding a constant to the outside of a function shifts the graph vertically. A positive constant of 4 moves every point 4 units upward, so the vertex moves from (0, 0) to (0, 4).
Multiply the coefficients 3 · (−2) = −6, then add the exponents for each base: x² · x = x³ and y · y³ = y⁴. The result is −6x³y⁴.
The flat fee is charged once regardless of distance, so it is the constant term. The per-mile charge multiplies the number of miles, giving C = 2m + 3.
The x-coordinate of the vertex of a parabola y = ax² + bx + c is given by x = −b/(2a). Here a = 1 and b = −8, so x = 8/2 = 4.
A quadratic has exactly one real solution when its discriminant b² − 4ac equals zero. Here k² − 4(1)(16) = 0, so k² = 64 and k = 8 or k = −8.
Evaluate the inner function first: g(2) = 2² − 1 = 3. Then apply f to that result: f(3) = 3(3) + 2 = 11.
Both equations describe lines with slope 2 but different y-intercepts, 1 and −3. Parallel distinct lines never intersect, so the system has no solution.
We need p · q = 20 with both positive, and b = p + q. The factor pairs are 1 and 20, 2 and 10, and 4 and 5, giving sums 21, 12, and 9. The smallest sum is 9.
Multiply the numerical coefficients: 4 times 3 equals 12. When multiplying powers with the same base, add the exponents: x² times x⁵ gives x raised to the seventh power. The product is therefore 12x⁷.
Substitute −3 for x: f(−3) = 4 − (−3)². Since (−3)² = 9, the result is 4 − 9 = −5. The subtraction must be done after squaring, and the square of a negative number is positive.
Subtract 5 from both sides to get −2x ≥ 6. Dividing both sides by −2 reverses the inequality symbol, so x ≤ −3. Remember that dividing or multiplying an inequality by a negative number flips the direction of the sign.
The expression is a difference of two squares: 4x² = (2x)² and 25 = 5². The pattern a² − b² factors as (a − b)(a + b), so 4x² − 25 = (2x − 5)(2x + 5).
Slope equals the change in y divided by the change in x: (13 − 5)/(6 − 2) = 8/4 = 2. The rise is 8 and the run is 4, giving a slope of 2.