Let's go through and solve each problem from the provided image:

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1. $64^{\frac{3}{2}}$

Using the property of exponents, $a^{\frac{m}{n}} = (\sqrt[n]{a})^m$:

$64^{\frac{3}{2}} = \left(\sqrt{64}\right)^3 = 8^3 = 512$

Answer: $512$

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2. $(-8)^{\frac{1}{3}}$

The cube root of $-8$ is:

$(-8)^{\frac{1}{3}} = \sqrt[3]{-8} = -2$

Answer: $-2$

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3. $4^{-\frac{3}{2}}$

Using the negative exponent rule and radical notation:

$4^{-\frac{3}{2}} = \frac{1}{4^{\frac{3}{2}}}$ $4^{\frac{3}{2}} = \left(\sqrt{4}\right)^3 = 2^3 = 8$ $4^{-\frac{3}{2}} = \frac{1}{8}$

Answer: $\frac{1}{8}$

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4. $x^{\frac{1}{2}} \cdot x^{\frac{1}{3}}$

Using the property of exponents, $a^m \cdot a^n = a^{m+n}$:

$x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{2} + \frac{1}{3}}$

Find the common denominator and add:

$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

$x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} = x^{\frac{5}{6}}$

Answer: $x^{\frac{5}{6}}$

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5. $\left(\frac{y^{\frac{1}{2}}}{y^{\frac{1}{4}}}\right)^{-1}$

First, simplify inside the parentheses using $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{y^{\frac{1}{2}}}{y^{\frac{1}{4}}} = y^{\frac{1}{2} - \frac{1}{4}}$

Find the common denominator:

$\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}$

So:

$\frac{y^{\frac{1}{2}}}{y^{\frac{1}{4}}} = y^{\frac{1}{4}}$

Now, apply the negative exponent outside the parentheses:

$\left(y^{\frac{1}{4}}\right)^{-1} = y^{-\frac{1}{4}}$

Answer: $y^{-\frac{1}{4}}$

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6. $\frac{x^{\frac{1}{4}}}{x^{-\frac{1}{2}}}$

Using $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{x^{\frac{1}{4}}}{x^{-\frac{1}{2}}} = x^{\frac{1}{4} - \left(-\frac{1}{2}\right)}$

Simplify the exponent:

$\frac{1}{4} - \left(-\frac{1}{2}\right) = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}$

$\frac{x^{\frac{1}{4}}}{x^{-\frac{1}{2}}} = x^{\frac{3}{4}}$

Answer: $x^{\frac{3}{4}}$

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Final Answers:

1. $512$
2. $-2$
3. $\frac{1}{8}$
4. $x^{\frac{5}{6}}$
5. $y^{-\frac{1}{4}}$
6. $x^{\frac{3}{4}}$

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Would you like further details on any specific step?
Here are five related questions to deepen your understanding:

1. How do fractional exponents relate to roots?
2. What happens when you multiply numbers with the same base but different exponents?
3. Why is the cube root of a negative number negative?
4. How do you simplify expressions with negative exponents?
5. Can fractional exponents be negative, and how do they work?

Tip: Always simplify exponents step-by-step to avoid errors, especially when dealing with fractions!