Select all expressions that can be differentiated using ONLY th \( \square \) A. \( \sqrt{x^{2}} \) \( \square \) B. \( x^{3} \cdot x^{9} \) \( \square \) C. \( 4^{x} \) \( \square \) D. \( 5 x^{-6} \) \( \square \) E. \( x^{3}+x^{9} \) \( \square \) F. \( x^{\sqrt{7}} \) \( \square \) G. \( x^{e} \)

**Step 1. Understand the “power rule” context.** We are asked to select those expressions that can be differentiated using only the power rule. The power rule states that if \[ f(x)= x^n, \] then \[ f'(x)= n x^{n-1}, \] provided \(n\) is a constant. **Step 2. Analyze each option.** **A. \(\sqrt{x^{2}}\)** We have: \[ \sqrt{x^2}= |x|. \] Because \(|x|\) is not differentiable at \(x=0\) and its expression is piecewise (being \(x\) for \(x\ge0\) and \(-x\) for \(x<0\)), it cannot be differentiated solely using the power rule. **B. \(x^{3}\cdot x^{9}\)** Using the laws of exponents: \[ x^3\cdot x^9 = x^{3+9}= x^{12}. \] This is a pure power function. The power rule applies directly. **C. \(4^{x}\)** This is an exponential function with a constant base \(4\) (which is not \(x\)). The derivative of an exponential function \(a^x\) involves the natural logarithm: \[ \frac{d}{dx} (a^x)= a^x \ln(a). \] Thus, it does not use solely the power rule. **D. \(5x^{-6}\)** This is directly a constant multiple of a power function: \[ 5x^{-6}. \] The power rule applies. **E. \(x^{3}+x^{9}\)** This is a sum of two power functions. The power rule applies term‐by-term. **F. \(x^{\sqrt{7}}\)** Here \(\sqrt{7}\) is a constant exponent. This is also a power function, so the power rule applies. **G. \(x^{e}\)** Even though \(e\) is an irrational number (and it is also the base of the natural logarithm), when it is an exponent in this context it is a constant. Therefore, this is a power function and the power rule applies. **Step 3. Select the options.** The expressions that are pure power functions and can be differentiated using only the power rule are: - **B. \(x^{3}\cdot x^{9}\)** - **D. \(5 x^{-6}\)** - **E. \(x^{3}+x^{9}\)** - **F. \(x^{\sqrt{7}}\)** - **G. \(x^{e}\)** **Final Answer:** Options B, D, E, F, and G.