Questions: 2x^3(x^2)^4 / 7x^(-3)(x^(-3))^(-3)

Solution Steps

To simplify the given expression, we need to apply the laws of exponents. Specifically, we will use the power of a power rule, the product of powers rule, and the quotient of powers rule. Simplify the expression step by step by combining the exponents.

The given expression is

\[ \frac{2 x^{3}\left(x^{2}\right)^{4}}{7 x^{(-3)}\left(x^{(-3)}\right)^{(-3)}} \]

First, apply the power of a power rule: \((a^m)^n = a^{m \cdot n}\).

\[ (x^2)^4 = x^{2 \cdot 4} = x^8 \]

\[ (x^{-3})^{-3} = x^{-3 \cdot (-3)} = x^9 \]

Substitute the simplified powers back into the expression:

\[ \frac{2 x^{3} \cdot x^{8}}{7 x^{-3} \cdot x^{9}} \]

Use the product of powers rule: \(a^m \cdot a^n = a^{m+n}\).

\[ 2 x^{3+8} = 2 x^{11} \]

\[ 7 x^{-3+9} = 7 x^{6} \]

Use the quotient of powers rule: \(\frac{a^m}{a^n} = a^{m-n}\).

\[ \frac{2 x^{11}}{7 x^{6}} = \frac{2}{7} x^{11-6} = \frac{2}{7} x^{5} \]

Final Answer