Simplify the Radical Expression (x^16y^20z^8)^(1/4)
The given problem is asking to perform a simplification on the expression provided, which is a radical expression in the form of a rational exponent. The expression contains variables x, y, and z raised to the power of 16, 20, and 8, respectively, and the entire expression is then raised to the power of 1/4. Simplifying the radical expression involves applying exponent rules to simplify the powers of the variables inside the radical.
$\left(\left(\right. x^{16} y^{20} z^{8} \left.\right)\right)^{\frac{1}{4}}$
Answer
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:
Power Rule: When you have a power raised to another power, you multiply the exponents. For example, $(a^m)^n = a^{m \cdot n}$.
Product Rule: When you have a product of terms raised to a power, you can apply the power to each term individually. For example, $(ab)^n = a^n b^n$.
Simplifying Exponents: When simplifying expressions with exponents, look for common factors that can be cancelled out to reduce the expression to its simplest form.
Radicals and Rational Exponents: A radical expression can be rewritten using rational (fractional) exponents. For example, $\sqrt[n]{a} = a^{\frac{1}{n}}$.
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.
