c) $$ \left(\frac{a^{\frac{3}{4}} b^{-2}}{a^{-1} b^{\frac{5}{2}}}\right)^{2} $$

Question

UpStudy AI · Accepted Answer

We start with the expression:
$$
\left(\frac{a^{\frac{3}{4}} b^{-2}}{a^{-1} b^{\frac{5}{2}}}\right)^{2}
$$

**Step 1: Simplify the fraction inside the parentheses**

Divide the powers of $$a$$ and $$b$$ separately.

For $$a$$:
$$
\frac{a^{\frac{3}{4}}}{a^{-1}} = a^{\frac{3}{4} - (-1)} = a^{\frac{3}{4} + 1} = a^{\frac{7}{4}}
$$

For $$b$$:
$$
\frac{b^{-2}}{b^{\frac{5}{2}}} = b^{-2 - \frac{5}{2}} = b^{-\left(2 + \frac{5}{2}\right)} = b^{-\frac{9}{2}}
$$

Thus, the expression inside becomes:
$$
a^{\frac{7}{4}}b^{-\frac{9}{2}}
$$

**Step 2: Raise the result to the power $$2$$**

We apply the exponent rule $$(x^m)^n = x^{m \cdot n}$$:
$$
\left(a^{\frac{7}{4}}b^{-\frac{9}{2}}\right)^2 = a^{\frac{7}{4}\cdot 2} \cdot b^{-\frac{9}{2}\cdot 2} = a^{\frac{7}{2}} \cdot b^{-9}
$$

**Final Answer:**
$$
a^{\frac{7}{2}}b^{-9}
$$
The simplified form of the expression is $$ a^{\frac{7}{2}}b^{-9} $$.

c) \( \left(\frac{a^{\frac{3}{4}} b^{-2}}{a^{-1} b^{\frac{5}{2}}}\right)^{2} \)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions