Question

1. a company manufactures photo cells. it uses the expression $(2x^{3})^{3}$ millimeters per second to calculate the maximum capacity of a photo cell with area $x^{3}$ square millimeters. use a property of exponents to simplify this expression.

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n=a^n\times b^n\). For \((2x^{3})^{3}\), we have \(a = 2\), \(b=x^{3}\), and \(n = 3\). So, \((2x^{3})^{3}=2^{3}\times(x^{3})^{3}\).

Step2: Calculate \(2^{3}\)

\(2^{3}=2\times2\times2 = 8\).

Step3: Apply the power of a power rule

The power of a power rule states that \((a^{m})^{n}=a^{m\times n}\). For \((x^{3})^{3}\), with \(a=x\), \(m = 3\), and \(n = 3\), we get \((x^{3})^{3}=x^{3\times3}=x^{9}\).

Step4: Combine the results

Since \(2^{3}=8\) and \((x^{3})^{3}=x^{9}\), then \(2^{3}\times(x^{3})^{3}=8x^{9}\).

Answer:

\(8x^{9}\)