\( A_{f} \) Ba St A Dc St ku Bl M .cloud-ext.icts.kuleuven.be/bba/ Consider the expression \( \left(\frac{b^{7} \cdot \sqrt{b}}{b \cdot \sqrt{b^{7}}}\right)^{3} \). This can be written as a single power of \( b \), namely \( b^{x} \) with \( x= \) \( \square \) This feedback is based on your last submitted answer. Submit your changed ans Submit answer Created using Numbas, developed by Newcastle University.

Let's simplify the expression \( \left(\frac{b^{7} \cdot \sqrt{b}}{b \cdot \sqrt{b^{7}}}\right)^{3} \). First, we rewrite \( \sqrt{b} \) as \( b^{1/2} \) and \( \sqrt{b^{7}} \) as \( b^{7/2} \). So the expression becomes: \[ \left(\frac{b^{7} \cdot b^{1/2}}{b \cdot b^{7/2}}\right)^{3} \] Now, combine the powers of \( b \) in the numerator and the denominator: \[ \frac{b^{7 + 1/2}}{b^{1 + 7/2}} = \frac{b^{7.5}}{b^{4.5}} \] Now, subtract the exponents of \( b \): \[ b^{7.5 - 4.5} = b^{3} \] Raise this to the power of 3: \[ (b^{3})^{3} = b^{9} \] Thus, \( x = 9 \). So the complete answer is \( x = \square \) which would be filled in with \( 9 \).