Question

Write your answer as a power or as a product of powers. $$ 4 x \cdot\left(x \cdot x^{3}\right)^{2} $$

Short answer

The simplified expression is \(4x^{9}\).

Step-by-step solution

Analyze the expression

The given expression \(4x \cdot (x \cdot x^{3})^{2} \) has an inner expression \(x \cdot x^{3}\) raised to the power of 2. We also have a 4x term that is being multiplied to this whole expression.

Simplify the inner expression

Make use of the multiplication rule when the bases are the same. Here, the base, x, is the same. Applying the rule \(x \cdot x^{3} = x^{1+3} = x^{4}\). So the expression now simplifies to: \(4x \cdot (x^{4})^{2}\).

Apply power of a power rule and simplify further

One can notice that there is an opportunity to apply the power of a power rule on \(x^{4})^{2}\). According to this rule, when an exponent is raised to another exponent, you multiply the exponents. Applying this rule \(x^{4})^{2} = x^{4*2} = x^{8}\). So, the expression now simplifies to: \(4x \cdot x^{8}\).

Apply multiplication rule with same bases once more

Finally, apply the multiplication rule for the same base once more on \(4x \cdot x^{8} = 4x^{1+8} = 4x^{9}\). Considering that the 4 is a constant, we must leave it as it is.