1. Let's simplify each algebraic expression step-by-step. 2. For addition and subtraction of like terms, combine coefficients and keep the variable part the same. 3. For multiplication, multiply coefficients and add exponents of like bases. 4. Simplify each expression: - $5x^2 + 2x^2 - 3x^2 = (5 + 2 - 3)x^2 = 4x^2$ - $(5x^2)(2x^2)(-3x^2) = 5 \times 2 \times (-3) \times x^{2+2+2} = -30x^6$ - $4x^3 + x^2 + 4x$ cannot be combined further as terms have different powers. - $(4x^3)(x^2)(4x) = 4 \times 1 \times 4 \times x^{3+2+1} = 16x^6$ - $-3x^3 + 5x^2 - 3x^3 = (-3 - 3)x^3 + 5x^2 = -6x^3 + 5x^2$ - $(-3x^3)(5x^2)(-3x^3) = (-3) \times 5 \times (-3) \times x^{3+2+3} = 45x^8$ - $3x + 2y$ cannot be combined. - $(3x)(2y) = 6xy$ - $7xy^2 - 2xy^2 = (7 - 2)xy^2 = 5xy^2$ - $(7xy^2)(-2xy^2) = 7 \times (-2) \times x^{1+1} \times y^{2+2} = -14x^2y^4$ - $7x^2y - 2xy^2$ cannot be combined. - $(7x^2y)(-2xy^2) = 7 \times (-2) \times x^{2+1} \times y^{1+2} = -14x^3y^3$ - $4x^2$ (with 18 crossed) means $\cancel{18}4x^2 = 4x^2$ - $4x^3 + x^2 + 4x$ no further simplification. - $45x^8$ is from above multiplication. - $-14x^3y^3$ from above. - $-30x^6$ from above. - $-14x^2y^4$ from above. - $6xy$ from above. - $-6x^3 + 5x^2$ from above. 5. Now for the expressions with $a$ and $b$: - $(3a)(a^2)(a^3) + (2a^2)(a^4) = 3a^{1+2+3} + 2a^{2+4} = 3a^6 + 2a^6 = 5a^6$ - $(a^4)(5a)(a^2) + (-4a^3)(2a^3)(a) = 5a^{4+1+2} + (-4 \times 2 \times 1)a^{3+3+1} = 5a^7 - 8a^7 = -3a^7$ - $(2a^3)(a^2)(3a^2) + (8a^2)(-a^2)(a) = 2 \times 3 \times a^{3+2+2} + 8 \times (-1) \times a^{2+2+1} = 6a^7 - 8a^5$ - $(5a^2)(2ab) + (a^2b)(3a) = 10a^{2+1}b + 3a^{2+1}b = 13a^3b$ - $(2ab^2)(-2a^2b^2) - (ab^3)(6a^2b) = -4a^{1+2}b^{2+2} - 6a^{1+2}b^{3+1} = -4a^3b^4 - 6a^3b^4 = -10a^3b^4$ - $(-a^2b)(ab^2)(a^2b^2) + (a^3b^2)(-a^2b^3) = -a^{2+1+2}b^{1+2+2} - a^{3+2}b^{2+3} = -a^5b^5 - a^5b^5 = -2a^5b^5$ - $(4a^2b^2)(-3b^3) - (2ab^2)(-6ab^3) = -12a^2b^{2+3} + 12a^{1+1}b^{2+3} = -12a^2b^5 + 12a^2b^5 = 0$ 6. Final answers: - $4x^2$ - $-30x^6$ - $4x^3 + x^2 + 4x$ - $16x^6$ - $-6x^3 + 5x^2$ - $45x^8$ - $3x + 2y$ - $6xy$ - $5xy^2$ - $-14x^2y^4$ - $7x^2y - 2xy^2$ - $-14x^3y^3$ - $4x^2$ - $4x^3 + x^2 + 4x$ - $45x^8$ - $-14x^3y^3$ - $-30x^6$ - $-14x^2y^4$ - $6xy$ - $-6x^3 + 5x^2$ - $5a^6$ - $-3a^7$ - $6a^7 - 8a^5$ - $13a^3b$ - $-10a^3b^4$ - $-2a^5b^5$ - $0$