Final answer:
To simplify (2x+3)^5, we use Pascal's triangle for the binomial expansion with the fifth row: 1, 5, 10, 10, 5, 1. We apply these coefficients to the terms of the binomial raised to the corresponding powers and multiply them accordingly to get the simplified form: 32x^5 + 720x^4 + 720x^3 + 1080x^2 + 810x + 243.
Step-by-step explanation:
To simplify (2x+3)^5 using Pascal's triangle, we use the binomial theorem and the corresponding row of Pascal's triangle for the exponent 5.
The row for exponent 5 is 1, 5, 10, 10, 5, 1. Now we expand the expression according to the binomial theorem.
The expansion is:
(2x+3)^5 = 1*(2x)^5 + 5*(2x)^4*(3) + 10*(2x)^3*(3^2) + 10*(2x)^2*(3^3) + 5*(2x)*(3^4) + 1*(3^5)
Which simplifies to:
(2x+3)^5 = 32x^5 + 240x^4*3 + 80x^3*9 + 40x^2*27 + 10x*81 + 243
And finally:
(2x+3)^5 = 32x^5 + 720x^4 + 720x^3 + 1080x^2 + 810x + 243