1. Simplify each expression by using the laws of exponents: $x^a \times x^b = x^{a+b}$ and $(x^a)^b = x^{a\times b}$. 2. For expressions involving multiplication of terms with the same base, add the exponents. For division, subtract the exponents. 3. For fractional powers, interpret them as roots: $x^{1/n} = \sqrt[n]{x}$. Examples from the set: 37. $5x^{-3} y^{3} \times 3x^{5} y^{3} = (5 \times 3) x^{-3+5} y^{3+3} = 15 x^{2} y^{6}$ 41. $(3p^4)^{-3} = 3^{-3} p^{4 \times -3} = \frac{1}{27} p^{-12} = \frac{1}{27 p^{12}}$ 51. $2^5 \times 2^{-3} = 2^{5-3} = 2^2 = 4$ 61. $81^{1/2} \times 27^{1/3} = (9^2)^{1/2} \times (3^3)^{1/3} = 9 \times 3 = 27$ ... and so on for all given problems. Because the user has requested many similar exponent problems, the same exponent rules apply to all. "slug": "exponent simplification", "subject": "algebra", "desmos": {"latex": "", "features": {"intercepts": false, "extrema": false}}, "q_count": 36