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Product Rule for Exponents
The product rule for exponents states that the multiplication of two identical bases, with differing exponents, results in the same base with the exponents added. In formula terms, x^a * x^b = x^(a + b). A variable example: x^3 * x^2 = x^(3 + 2) = x^5. An integer example: 3^3 * 3^4 = 3^(3 + 4) = 3^7, which then could be calculated to 2,187.
Quotient Rule for Exponents
The quotient rule for exponents states that in the division of like bases with differing exponents, the result is the base raised to the subtraction of the exponents. In formula form: (x^a) / (x^b) = x^(a - b). A variable example: (x^5) / (x^3) = x^(5 - 3) = x^2. An integer example: (2^8) / (2^6) = 2^(8 - 6) = 2^2, which equals 4.
Power Rule for Exponents
The power rule for exponents applies when the base and an exponent are inside parentheses and another exponent is applied to the exterior. The formula states that (x^m)^n = x^(m * n). A variable example: (x^3)^2 = x^(3 * 2) = x^6. An integer example: (2^3)^2 = 2^(3 * 2) = 2^6, which equals 64.
Power of a Product Rule
The power of a product rule applies to differing bases multiplied within a set of parentheses and raised to an exterior exponent. The formula states that (xy)^a = x^a * y^a. A variable example: (xy)^7 = x^7 * y^7. An integer with variable example: (2x)^3 = 2^3 * x^3, which can be simplified to 8x^3.
Power of a Quotient Rule
The power of a quotient rule states that, for a division of differing bases, (x / y)^a = (x^a) / (y^a). A variable example of the rule: (x / y)^10 = (x^10) / (y^10). Note that the exponents can't be cancelled out because the bases are different. An integer with variable example: (x / 5)^2 = (x^2) / (5^2) = (x^2) / 25.