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# Properties of Exponents

The table below lists several properties of exponents. Here, m and n are real numbers.

 Name Property Example Multiplication Property xm Â· xm = x m + n 54 Â· 52 = x 4 + 2 = 56 Division Property Power of a Power Property Power of a Product Property (xy)n = xn yn (2x)3 = 23 x3 = 8x3 Power of a Quotient Property Zero Power Property x0 = 1, x ≠ 0 170 = 1 Negative Exponent

Example 1

Simplify: 2xy4(3x5y)(x3y5)2

 We can drop parentheses around the quantity (3x5y) since it has an exponent of 1. Solution 2xy4(3x5y)(x3y5)2 That is, (3x5y) = (3x5y)1 = 3x5y. Use the Power of a Product Property. = 2xy4(3x5y)(x3Â·2y5Â·2) Simplify. = 2xy4(3x5y)(x6y10) = 2xy4 Â· 3x5y Â· x6y10 Recall: x = x1 and  y = y1 Multiply the constants, 2 and 3. = 6xy4 Â· x5y Â· x6y10 Use the Multiplication Property of Exponents. = 6x1+5+6 y4+1+10 So, 6xy4 = 6x1y4 and x5y = x5y1. Simplify. = 6x12 y15 So, the result is 6x12y15.