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√18 simplifies to 3√2, and √2 cannot be written as a ratio of two integers, so √18 is irrational. The other three values are rational: √49 = 7, the decimal 0.272727… repeats a fixed block, and −7/3 is already a quotient of integers.
When multiplying powers with the same base, add the exponents: 3 + (−5) = −2, so the product is 2^−2. A negative exponent means the reciprocal of the positive power, giving 1/2^2 = 1/4.
Scientific notation requires a coefficient between 1 and 10 multiplied by a power of ten. Moving the decimal point in 0.00047 four places to the right gives 4.7, and since the original number is less than 1 the exponent is negative four.
The negative exponent gives the reciprocal, so 27^−2/3 = 1/27^2/3. Since 27^1/3 = 3, the expression equals 1/3^2 = 1/9.
Write each radical in simplest form using the largest perfect-square factor: √50 = 5√2, √18 = 3√2, and √8 = 2√2. Combining like radicals gives 5√2 + 3√2 − 2√2 = 6√2.
Distribute each term: 3(1) + 3(−4i) + 2i(1) + 2i(−4i) = 3 − 12i + 2i − 8i^2. Because i^2 = −1, the term −8i^2 equals +8, so the real part is 3 + 8 = 11 and the imaginary part is −12i + 2i = −10i.
Multiply by the conversion factors so that unwanted units cancel: 60 mi/h × 5,280 ft/1 mi × 1 h/3,600 s. This gives 316,800 feet divided by 3,600 seconds, which equals 88 feet per second.
A power of a product lets each factor be raised separately: 8^2/3 · (x^6)^2/3. Since 8^2/3 = (8^1/3)^2 = 2^2 = 4 and (x^6)^2/3 = x^4, the simplified form is 4x^4.
Rationalize by multiplying numerator and denominator by the conjugate 3 + √5. The denominator becomes 9 − 5 = 4 and the numerator becomes 4(3 + √5), so the fraction reduces to 3 + √5.
If r + s equaled a rational number q, then s = q − r would be rational, contradicting that s is irrational, so the sum must be irrational. The other statements fail by counterexample: √2 · √2 = 2, √9 = 3, and √2 is real but neither rational nor an integer.
Multiply the coefficients: 2.5 × 4.0 = 10. Then add the exponents: 10^4 × 10^−7 = 10^−3. Because 10 × 10^−3 = 1.0 × 10^−2, the product in scientific notation is 1.0 × 10^−2.
A negative exponent means to take the reciprocal of the base and then apply the positive exponent: (2/3)^−2 = (3/2)^2. Squaring gives 9/4. The result is positive because the exponent 2 is even.
The product has the form (a+b)(a−b)=a^2−b^2 with a=√7 and b=√3. Thus (√7)^2 − (√3)^2 = 7 − 3 = 4. The middle terms cancel.
Factor each radicand using perfect cubes: 54 = 27·2, so ∛54 = 3∛2; and 16 = 8·2, so ∛16 = 2∛2. Adding like radical terms gives 3∛2 + 2∛2 = 5∛2.
Powers of i repeat every 4: i^1=i, i^2=−1, i^3=−i, i^4=1. Since 38 mod 4 = 2, i^38 = −1; since 47 mod 4 = 3, i^47 = −i. Therefore the sum is −1 − i.
Raise both sides to the reciprocal power 3/2: x = 16^(3/2). Since 16^(1/2)=4 and 4^3=64, the positive solution is x=64.
The area in square yards is 4.5 × 3.0 = 13.5 square yards. Because 1 yard = 3 feet, 1 square yard = 9 square feet. Thus 13.5 × 9 = 121.5 square feet.
Write 72 as 36·2 and split the variables: √(72x^5y^4) = √36 · √(x^4) · √(y^4) · √(2x). This equals 6 · x^2 · y^2 · √(2x), or 6x^2y^2√(2x).
Multiply numerator and denominator by the conjugate 2+i. The numerator becomes (4+3i)(2+i)=8+4i+6i+3i^2=5+10i, and the denominator becomes 2^2+1^2=5. Dividing gives 1+2i.
Convert grams to kilograms by dividing by 1,000, and convert cubic centimeters to cubic meters using 1 cm^3 = 10^−6 m^3. Thus 2.7 g/cm^3 = 2.7 × (0.001 kg)/(10^−6 m^3) = 2,700 kg/m^3.
Add complex numbers by combining real parts and imaginary parts separately. The real part is 7 + (−2) = 5, and the imaginary part is −3 + 8 = 5, giving 5 + 5i.
In an expression x^(m/n), the denominator n is the index of the radical and the numerator m is the exponent inside the radical. Therefore x^(3/4) equals the fourth root of x cubed, written ∜(x^3).
For positive numbers in scientific notation, the power of 10 with the greatest exponent is the largest value. Here the exponents are −3, −4, −2, and −3; the greatest exponent is −2, so 3.0 × 10^−2 is greatest.
This product has the form (a − bi)(a + bi), a difference of squares. It equals a^2 − (bi)^2 = 4^2 − (7i)^2 = 16 + 49 = 65, because i^2 = −1.
Apply the exponent −2 to both factors: (2x^3)^−2 = 2^−2 (x^3)^−2 = 1/4 · x^−6. Rewriting with positive exponents gives 1/(4x^6).
To rationalize a cube-root denominator, multiply by ∛4 so that ∛2 · ∛4 = ∛8 = 2. Then 6∛4 / 2 simplifies to 3∛4.
Divide the coefficients and subtract the exponent of the denominator from the exponent of the numerator. 8.4 ÷ 2.1 = 4.0 and 10^6 ÷ 10^2 = 10^4, so the result is 4.0 × 10^4.
Square both sides of √(3x + 4) = 5 to get 3x + 4 = 25. Subtract 4 and divide by 3 to obtain x = 7, and checking gives √25 = 5.
Subtract the complex numbers: (3 + 4i) − (−1 + i) = 4 + 3i. The distance is the modulus √(4^2 + 3^2) = √25 = 5.
First square the complex number: (1 + i)^2 = 1 + 2i + i^2 = 2i. Then square again: (2i)^2 = 4i^2 = −4, so the result is −4 + 0i.
The product of square roots rule allows √12 · √3 = √(12 · 3) = √36. Since 36 is a perfect square, √36 = 6.
Scientific notation has the form a × 10^n where 1 ≤ |a| < 10. The decimal 0.000032 moves the decimal point five places to get 3.2, so the exponent is −5.
A rational exponent m/n means take the nth root and then raise to the m power. Here 27^(2/3) = (∛27)^2 = 3^2 = 9.
When multiplying powers with the same base, add the exponents: 1/2 + 1/3 = 5/6. When dividing, subtract the exponent: 5/6 − 1/6 = 4/6 = 2/3, so the expression equals x^(2/3).
Simplify each radical first: √48 = √(16 · 3) = 4√3 and √27 = √(9 · 3) = 3√3. Adding like radicals gives 4√3 + 3√3 = 7√3.
Use the distributive property: (2 + 3i)(1 − 4i) = 2 − 8i + 3i − 12i^2. Since i^2 = −1, the last term is +12, giving 14 − 5i.
Multiply numerator and denominator by the conjugate 3 + √5. The denominator becomes 9 − 5 = 4, and the numerator 4(3 + √5) divided by 4 simplifies to 3 + √5.
Convert miles to feet and hours to seconds: 60 miles/hour × 5280 feet/mile × 1 hour/3600 seconds. The computation gives 88 feet per second.
Expand: (a + 2i)(3 − i) = 3a − ai + 6i − 2i^2 = (3a + 2) + (6 − a)i. For the result to be real, the imaginary part must be zero, so 6 − a = 0 and a = 6.
Apply the exponent to each factor: (16x^8)^(3/4) = 16^(3/4) · (x^8)^(3/4). Since 16^(3/4) = (∜16)^3 = 8 and (x^8)^(3/4) = x^6, the result is 8x^6.
Multiply the coefficients: 3 · (−2) = −6, which is negative. When multiplying powers with the same base, add the exponents: x² · x = x³ and y · y³ = y⁴. Combining the coefficient with both variable parts gives −6x³y⁴.
Slope is the change in y divided by the change in x: (−7 − 5)/(4 − (−2)) = −12/6 = −2. Subtracting the coordinates in the same order in the numerator and denominator keeps the sign correct, and the line falls from left to right.
Substitute −2 wherever x appears: 2(−2)² − 3(−2) + 1 = 2(4) + 6 + 1 = 15. Squaring before multiplying by 2, and recognizing that −3 times −2 is positive 6, are the key steps.
Let s be the number of student tickets; then 300 − s adult tickets were sold. The revenue equation is 12(300 − s) + 7s = 2860, which simplifies to 3600 − 5s = 2860, so 5s = 740 and s = 148. Checking: 148 student tickets and 152 adult tickets give 1,036 + 1,824 = $2,860.
Factor by finding two numbers whose product is −24 and whose sum is −5, namely −8 and 3. Then g(x) = (x − 8)(x + 3), so the zeros occur where x − 8 = 0 or x + 3 = 0, giving x = 8 and x = −3.
Distribute on the left to get 4x − 8 ≤ 2x + 6, then subtract 2x from both sides and add 8 to both sides to obtain 2x ≤ 14. Dividing by positive 2 gives x ≤ 7, and the inequality symbol flips only when multiplying or dividing by a negative number, which never happens here.
The constant rate is (50 − 62)/(10 − 4) = −12/6 = −2 gallons per minute. Using the point (4, 62) in point-slope form gives G = 62 − 2(t − 4), which simplifies to G = 70 − 2t. Substituting t = 10 confirms G = 50.
Because the population doubles once every 3 hours, the number of doubling periods in t hours is t/3. Each doubling multiplies the population by 2, so the population is 500 · 2^(t/3); at t = 3 this gives 500 · 2 = 1,000, as expected.
Setting the two expressions for y equal gives x + 1 = x² − 1, so x² − x − 2 = 0, which factors as (x − 2)(x + 1) = 0. The x-values 2 and −1 give y-values 3 and 0, so the intersection points are (2, 3) and (−1, 0), and each satisfies both equations.
Set −16t² + 48t + 4 = 36, which becomes −16t² + 48t − 32 = 0 and simplifies to t² − 3t + 2 = 0. Factoring gives (t − 1)(t − 2) = 0, so the ball is at exactly 36 feet at t = 1 and t = 2 and above that height between them. The length of the interval is 2 − 1 = 1 second.