Question

simplify the following expression.
$(-5b^{9})^{2}(2b^{5}w^{9})^{3}$
show your work here
hint: to add an exponent (x^n), type \exponent\ or press \^\

Explanation:

Step1: Expand each term via exponent rule

First, apply the power of a product rule \((xy)^n = x^n y^n\) and power rule \((x^m)^n = x^{mn}\) to each factor:
For \((-5b^8)^2\):
\((-5)^2 \cdot (b^8)^2 = 25 \cdot b^{8 \times 2} = 25b^{16}\)
For \((2b^5 w^9)^3\):
\(2^3 \cdot (b^5)^3 \cdot (w^9)^3 = 8 \cdot b^{5 \times 3} \cdot w^{9 \times 3} = 8b^{15}w^{27}\)

Step2: Multiply the simplified terms

Multiply the coefficients and apply the product rule \(x^m \cdot x^n = x^{m+n}\) to like variables:
\(25b^{16} \times 8b^{15}w^{27} = (25 \times 8) \cdot (b^{16+15}) \cdot w^{27}\)

Step3: Compute final values

Calculate the coefficient sum and variable exponents:
\(25 \times 8 = 200\), \(b^{16+15} = b^{31}\)

Answer:

\(200b^{31}w^{27}\)