Question

1. \\(\left( 4b^3 \cdot 2b^2 \cdot b \

ight)^3\\)\

1. \\(\left( 3d^2 \cdot d^2 \

ight)^3\\)

Explanation:

Problem 6:

Step1: Multiply coefficients and combine like terms

First, multiply the coefficients \(4\) and \(2\), and combine the \(b\)-terms using the rule \(a^m \cdot a^n = a^{m + n}\). So, \(4b^3 \cdot 2b^2 \cdot b = (4 \cdot 2) \cdot b^{3 + 2 + 1}= 8b^6\).

Step2: Apply the power of a power rule

Now, raise \(8b^6\) to the power of \(3\) using \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\). So, \((8b^6)^3 = 8^3 \cdot (b^6)^3 = 512b^{18}\).

Step1: Combine like terms

First, combine the \(d\)-terms using \(a^m \cdot a^n = a^{m + n}\). So, \(3d^2 \cdot d^2 = 3d^{2 + 2}= 3d^4\).

Step2: Apply the power of a power rule

Now, raise \(3d^4\) to the power of \(3\) using \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\). So, \((3d^4)^3 = 3^3 \cdot (d^4)^3 = 27d^{12}\).

Answer:

\(512b^{18}\)

Problem 7: