Algebra Class: Variation Concepts

A relationship where one variable increases (or decreases) in proportion to another. As one variable changes, the other changes in the same direction.

y = 5x

A relationship where one variable increases as another decreases, or vice-versa, and their product stays constant.

r = k/t

y = kx

The equation that represents a direct variation between two variables, y and x, is y = kx, where k is the constant of variation.

The constant of variation is the factor that relates the two variables in direct variation. It's represented by 'k' in the equation y = kx.

The equation that represents an inverse variation between two variables x and y is y = k/x, where k is the constant of variation.

A relationship where one variable varies jointly with two or more other variables. The equation is typically of the form z = kxy, where k is the constant of variation.

Given a value of x and its corresponding value of y, find the constant of variation (k) by rearranging the direct variation equation: k = y/x.

A relationship involving both direct and inverse variation.

The constant value that links variables in a variation equation.

Expressing the relationship between variables using 'k' and exponents.

A problem where one quantity increases proportionally to another. As one variable goes up, the other goes up at the same rate.

A problem where one quantity decreases as another increases, and their product stays constant.

A problem where one quantity is directly proportional to the product of two or more other quantities.

1. Find the constant of proportionality (k) by dividing the first value of y by the first value of x. 2. Use the constant (k) to find the unknown value.

1. Find the constant of proportionality (k) by multiplying the initial values of x and y together. 2. Use the constant (k) to find the unknown value.

Always equals 1. For example, 5⁰ = 1, 100⁰ = 1, and even (x+y)⁰ = 1.

Results in 1, regardless of the value of x.

Evaluates to 1. Even if the base is an expression like (2a + 3b)⁰, it still equals 1.

Flip the base and make the exponent positive.

Equals 1/3.

Any non-zero number raised to the power of zero equals 1.

A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent.

A fractional exponent represents a root operation. The denominator of the fraction indicates the type of root, and the numerator indicates the power to which the base is raised.

To simplify expressions with fractional exponents, apply the rules of exponents for multiplication and division: (a^m)^n = a^(m*n) and a^m/a^n = a^(m-n).

To express a fraction in exponential form, write both the numerator and denominator as products of their prime factors. Then apply exponent rules to simplify.

This expression represents the cube of m squared multiplied by n squared. It is not equivalent to m2/3, which represents the cube root of m squared.

This expression represents the cube of m squared. Any number raised to the power of zero equals 1, so n0 equals 1. This is not equivalent to m2/3.

This expression represents the square of the cube root of 'm' (m1/3). n0 equals 1. This is not equivalent to m2/3.

This expression represents the square of the cube root of n (n1/3), as m raised to the power of 0 is 1. This is not equivalent to m2/3.

The square root of 64 is the number that, when multiplied by itself, equals 64. This number is 8.

When multiplying square roots, you multiply the numbers inside the radical sign. For example, the product of (√8)(√4) is √(8*4) = √32.

Radical terms are considered similar if they have the same radicand (the number under the radical sign). For example, √5, 2√5, and 3√5 are similar.

To simplify a radical, you find the largest perfect square that divides the radicand. Then, you take the square root of that perfect square and place it outside the radical.

The largest perfect square that divides 32 is 16. So √32 = √(16*2) = √16 * √2 = 4√2.

When multiplying exponents with the same base, add the powers together. (x^m)(x^n) = x^(m+n)

When raising an exponent to another power, multiply the powers together. (x^m)^n = x^(m*n)

Apply the power to a power rule: (4^(1/3))^2 = 4^(2/3). Then, rewrite as a radical: 4^(2/3) = ³√(4²)= ³√16. Finally, simplify the radical: ³√16 = ³√(8*2) = 2³√2.

Apply the power to a power rule: (5^(5/3))^2 = 5^(10/3). Convert to radical form: 5^(10/3) = ³√(5^10) = ³√(5^9 * 5) = 5³ ³√5 = 125³√5.

Apply the product rule of exponents: (5^(5/3))(5^(7/3)) = 5^(5/3 + 7/3) = 5^(12/3) = 5^4.

Combining square roots involves simplifying them by finding the largest perfect square factor and taking its square root. For example, √12 can be simplified as √(4 x 3) = 2√3.

When multiplying square roots, multiply the numbers inside the radicals. For example, (√4)(√8) = √(4 x 8) = √32, which can be simplified further.

Dividing square roots involves dividing the numbers inside the radicals. For example, √18 / √2 = √(18/2) = √9 = 3.

Raising a square root to a power involves applying the power to both the number inside the radical and the square root itself. For example, (√12)³ = (√12)(√12)(√12) = √(12 x 12 x 12) = √1728.