Simplify the Radical Expression (x^16y^20z^8)^(1/4)

Solution:

Step 1: Distribute the exponent across the terms inside the radical

Use the property $(ab{)}^n=a^n{b}^n$ to separate the exponent for each variable.

Step 1.1: Separate the exponent for $x^{16}{y}^{20}$

$(x^{16}{y}^{20}{)}^{\frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 1.2: Further separate the exponent for $x^{16}$ and $y^{20}$

$(x^{16}{)}^{\frac{1}{4}}(y^{20}{)}^{\frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 2: Simplify the exponent for $x^{16}$

Step 2.1: Apply the power of a power rule

$x^{16\cdot \frac{1}{4}}(y^{20}{)}^{\frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 2.2: Simplify the exponent by multiplying

Step 2.2.1: Simplify $16\cdot \frac{1}{4}$

$x^{4\cdot 4\cdot \frac{1}{4}}(y^{20}{)}^{\frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 2.2.2: Reduce the exponent by cancelling out the common factor

$x^4(y^{20}{)}^{\frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 2.2.3: Write down the simplified expression

$x^4(y^{20}{)}^{\frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 3: Simplify the exponent for $y^{20}$

Step 3.1: Apply the power of a power rule

$x^4{y}^{20\cdot \frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 3.2: Simplify the exponent by multiplying

Step 3.2.1: Simplify $20\cdot \frac{1}{4}$

$x^4{y}^{5\cdot 4\cdot \frac{1}{4}}(z^8{)}^{\frac{1}{4}}$

Step 3.2.2: Reduce the exponent by cancelling out the common factor

$x^4{y}^5(z^8{)}^{\frac{1}{4}}$

Step 3.2.3: Write down the simplified expression

$x^4{y}^5(z^8{)}^{\frac{1}{4}}$

Step 4: Simplify the exponent for $z^8$

Step 4.1: Apply the power of a power rule

$x^4{y}^5{z}^{8\cdot \frac{1}{4}}$

Step 4.2: Simplify the exponent by multiplying

Step 4.2.1: Simplify $8\cdot \frac{1}{4}$

$x^4{y}^5{z}^{2\cdot 4\cdot \frac{1}{4}}$

Step 4.2.2: Reduce the exponent by cancelling out the common factor

$x^4{y}^5{z}^2$

Step 4.2.3: Write down the simplified expression

$x^4{y}^5{z}^2$

Knowledge Notes:

1. Power Rule: When you have a power raised to another power, you multiply the exponents. For example, $(a^m{)}^n=a^{m\cdot n}$.

2. Product Rule: When you have a product of terms raised to a power, you can apply the power to each term individually. For example, $(ab{)}^n=a^n{b}^n$.

3. Simplifying Exponents: When simplifying expressions with exponents, look for common factors that can be cancelled out to reduce the expression to its simplest form.

4. Radicals and Rational Exponents: A radical expression can be rewritten using rational (fractional) exponents. For example, $\sqrt[n]{a}=a^{\frac{1}{n}}$.

5. Combining the Rules: In problems involving both product rule and power rule, it's important to apply the rules systematically to simplify the expression step by step.