$$ A_{f} $$ 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.

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UpStudy AI · Accepted Answer

We start with the expression

$$
\left(\frac{b^{7} \cdot \sqrt{b}}{b \cdot \sqrt{b^{7}}}\right)^{3}.
$$

**Step 1: Rewrite the square roots in exponential form.**

Recall that $$\sqrt{b} = b^{\frac{1}{2}}$$ and $$\sqrt{b^{7}} = b^{\frac{7}{2}}$$.

Thus, the expression becomes:

$$
\left(\frac{b^{7} \cdot b^{\frac{1}{2}}}{b \cdot b^{\frac{7}{2}}}\right)^{3}.
$$

**Step 2: Simplify the exponents in the numerator and denominator.**

For the numerator:

$$
b^{7} \cdot b^{\frac{1}{2}} = b^{7 + \frac{1}{2}} = b^{\frac{14}{2} + \frac{1}{2}} = b^{\frac{15}{2}}.
$$

For the denominator:

$$
b \cdot b^{\frac{7}{2}} = b^{1 + \frac{7}{2}} = b^{\frac{2}{2} + \frac{7}{2}} = b^{\frac{9}{2}}.
$$

Thus, the expression inside the parentheses reduces to:

$$
\frac{b^{\frac{15}{2}}}{b^{\frac{9}{2}}} = b^{\frac{15}{2} - \frac{9}{2}} = b^{\frac{6}{2}} = b^{3}.
$$

**Step 3: Raise the simplified expression to the power of 3.**

$$
\left(b^3\right)^3 = b^{3 \cdot 3} = b^9.
$$

Therefore, the given expression can be written as:

$$
b^9,
$$

which means $$x = 9$$.
$$ x = 9 $$

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