Expression Comparison: Determining Greater Value When x > 1

To find which expression has the greatest value given $x > 1$, we will apply exponent rules:

• Expression (1): $x^2 \times x^9$
• Expression (2): $x^2 \times x^3$
• Expression (3): $x^{10} \times x^{-7}$
• Expression (4): $x \times x$

Now, let's simplify each expression:

• Expression (1): Using $x^a \times x^b = x^{a+b}$, we get $x^2 \times x^9 = x^{2+9} = x^{11}$.
• Expression (2): Similarly, $x^2 \times x^3 = x^{2+3} = x^5$.
• Expression (3): Thus, $x^{10} \times x^{-7} = x^{10-7} = x^3$.
• Expression (4): It becomes $x \times x = x^{1+1} = x^2$.

Now, compare the powers: $11, 5, 3,$ and $2$. Since $x > 1$, the greater the exponent, the greater the value of the expression. Thus, the expression with the largest power of $x$ is $x^{11}$ from expression (1).

Therefore, the expression with the largest value is $x^2 \times x^9$.