Let's simplify each given radical expression step by step.

45.

(a) $\sqrt[4]{x^4}$
Using the property of radicals $\sqrt[n]{a^n} = a$:
$\sqrt[4]{x^4} = x$

(b) $\sqrt[4]{16x^8}$
Break it into separate radicals:
[ \sqrt[4]{16x^8} = \sqrt[4]{16} \cdot \sqrt[4]{x^8} ]
$\sqrt[4]{16} = 2$ because $2^4 = 16$, and $\sqrt[4]{x^8} = x^2$ (divide the exponent by 4).
$\sqrt[4]{16x^8} = 2x^2$

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46.

(a) $\sqrt[5]{x^{10}}$
Using the property $\sqrt[n]{x^m} = x^{m/n}$:
$\sqrt[5]{x^{10}} = x^{10/5} = x^2$

(b) $\sqrt[3]{x^3y^6}$
Separate terms:
[ \sqrt[3]{x^3y^6} = \sqrt[3]{x^3} \cdot \sqrt[3]{y^6} ]
$\sqrt[3]{x^3} = x$ and $\sqrt[3]{y^6} = y^2$:
$\sqrt[3]{x^3y^6} = xy^2$

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47.

(a) $\sqrt[6]{64a^6b^7}$
Break into separate radicals:
[ \sqrt[6]{64a^6b^7} = \sqrt[6]{64} \cdot \sqrt[6]{a^6} \cdot \sqrt[6]{b^7} ]
$\sqrt[6]{64} = 2$ because $2^6 = 64$, $\sqrt[6]{a^6} = a$, and $\sqrt[6]{b^7} = b^{7/6}$:
$\sqrt[6]{64a^6b^7} = 2ab^{7/6}$

(b) $\sqrt[3]{a^2b} \cdot \sqrt[3]{64a^4b}$
Combine under one cube root:
[ \sqrt[3]{a^2b \cdot 64a^4b} = \sqrt[3]{64a^{2+4}b^{1+1}} = \sqrt[3]{64a^6b^2} ]
Separate and simplify:
[ \sqrt[3]{64a^6b^2} = \sqrt[3]{64} \cdot \sqrt[3]{a^6} \cdot \sqrt[3]{b^2} ]
$\sqrt[3]{64} = 4$, $\sqrt[3]{a^6} = a^2$, and $\sqrt[3]{b^2} = b^{2/3}$:
$\sqrt[3]{64a^6b^2} = 4a^2b^{2/3}$

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48.

(a) $\sqrt[4]{x^4y^2z^2}$
Separate terms:
[ \sqrt[4]{x^4y^2z^2} = \sqrt[4]{x^4} \cdot \sqrt[4]{y^2} \cdot \sqrt[4]{z^2} ]
$\sqrt[4]{x^4} = x$, $\sqrt[4]{y^2} = y^{2/4} = y^{1/2}$, and $\sqrt[4]{z^2} = z^{2/4} = z^{1/2}$:
$\sqrt[4]{x^4y^2z^2} = xy^{1/2}z^{1/2}$

(b) $\sqrt[3]{\sqrt[3]{64x^6}}$
Simplify the inner cube root first:
[ \sqrt[3]{64x^6} = \sqrt[3]{64} \cdot \sqrt[3]{x^6} = 4x^2 ]
Now take the cube root again:
[ \sqrt[3]{4x^2} = \sqrt[3]{4} \cdot \sqrt[3]{x^2} ]
Final result:
[ \sqrt[3]{\sqrt[3]{64x^6}} = \sqrt[3]{4}x^{2/3} ]

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Questions to Explore Further:

1. What are the key properties of radicals that simplify fractional exponents?
2. How do higher roots (e.g., 4th or 6th roots) differ from square roots in simplification?
3. What happens when radical expressions include negative exponents?
4. How do you simplify radicals with coefficients outside the radical?
5. How can radical expressions be rationalized when involving denominators?

Tip: Always rewrite radicals involving even roots as fractional exponents when stuck—it simplifies the process.