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Solve for x: √x + 3 - y = 0.
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x = y^2 - 3 (move terms and square both sides).
Explain how to solve the equation: y/x + a = b.
x = y/(b - a) (move terms and multiply by x).
Solve using the quadratic formula: 2x^2 = 3x + 5.
Rearrange to 2x^2 - 3x - 5 = 0, then x = (-(-3) ± √((-3)^2 - 4(2)(-5))) / 2(2), so x = 2.5 or x = -1.
Rearrange and solve: ax - y = 2y.
x = 3y/a (move terms and divide by a).
What is the process for dividing fractions? Give an example.
To divide fractions, multiply by the reciprocal of the divisor. Example: a/b / c/d = a/b * d/c = ad/bc.
What is the product rule for exponents and provide an example?
The product rule for exponents states that x^m * x^n = x^(m+n). Example: 3^3 * 3^2 = 3^(3+2) = 3^5 = 243.
How do you calculate the square root of a number using fractional exponents? Provide an example.
Fractional exponents can represent roots, such that x^(1/2) is the square root of x. Example: 4^(1/2) = √4 = 2.
Explain the quotient rule for exponents with an example.
The quotient rule for exponents states that x^m / x^n = x^(m-n). Example: 3^3 / 3^2 = 3^(3-2) = 3^1 = 3.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
What is the basic principle for solving equations and provide an example.
The basic principle for solving equations is to perform the same operation on both sides of the equation. Example: y = x/5 → x = 5y (multiply by 5).
Explain the negative exponent rule with an example.
The negative exponent rule states that x^-n = 1/x^n. Example: 2^-1 = 1/2 = 0.5.
Discuss the distributive property in the context of solving: a(x - y) = b(x + y).
Distribute to get ax - ay = bx + by, combine like terms to get ax - bx = ay + by, factor to get x(a - b) = y(a + b), then solve: x = y(a + b)/(a - b).
What is the distributive rule for exponents? Include an example.
The distributive rule for exponents states that (xy)^m = x^m * y^m. Example: (3*2)^4 = 3^4 * 2^4 = 81 * 16 = 1296.
How do you multiply fractions? Provide an example.
To multiply fractions, multiply the numerators together and the denominators together. Example: a/b * c/d = ac/bd.
Describe the steps to add or subtract fractions with different denominators. Give an example.
To add or subtract fractions, find a common denominator and then combine the numerators. Example: a/b ± c/d = (ad ± bc)/bd.
Simplify and solve: a(x + b) = c.
x = c/a - b (divide by a and move terms).
Describe the power rule for exponents and give an example.
The power rule for exponents states that (x^m)^n = x^(mn). Example: 2^2^3 = 2^6 = 64.
What is the importance of learning to rearrange equations before plugging in numbers?
Learning to rearrange equations is crucial because it allows for a clear understanding of relationships between variables and simplifies the process of solving for unknowns.
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