Simplify ( fifth root of 243a^12)/( fifth root of b^10)
This question requires simplifying an algebraic expression involving fifth roots. The expression is a division of two fifth root terms: the fifth root of 243 times a to the power of 12, divided by the fifth root of b to the power of 10. The task is to simplify this expression by using the properties of exponents and roots to combine the two separate fifth roots into one simplified expression, if possible.
$\frac{\sqrt[5]{243 a^{12}}}{\sqrt[5]{b^{10}}}$
Answer
Solution:
Simplification Process
Step 1: Simplify the Numerator
Step 1.1: Express $243a^{12}$ in a different form
- Express $243$ as $3^5$: $\frac{\sqrt[5]{3^5 a^{12}}}{\sqrt[5]{b^{10}}}$
- Separate $a^{12}$ into $a^{10} \cdot a^2$: $\frac{\sqrt[5]{3^5 (a^{10} a^2)}}{\sqrt[5]{b^{10}}}$
- Express $a^{10}$ as $(a^2)^5$: $\frac{\sqrt[5]{3^5 ((a^2)^5 a^2)}}{\sqrt[5]{b^{10}}}$
- Combine $3^5$ and $(a^2)^5$: $\frac{\sqrt[5]{(3 a^2)^5 a^2}}{\sqrt[5]{b^{10}}}$
Step 1.2: Extract terms from the radical
- Extract $(3a^2)^5$ as $3a^2$: $\frac{3 a^2 \sqrt[5]{a^2}}{\sqrt[5]{b^{10}}}$
Step 2: Simplify the Denominator
Step 2.1: Rewrite $b^{10}$ as $(b^2)^5$
- Rewrite the denominator: $\frac{3 a^2 \sqrt[5]{a^2}}{\sqrt[5]{(b^2)^5}}$
Step 2.2: Extract terms from the radical, assuming all are real numbers
- Extract $(b^2)^5$ as $b^2$: $\frac{3 a^2 \sqrt[5]{a^2}}{b^2}$
Knowledge Notes:
The problem involves simplifying a radical expression with a fifth root. Here are the relevant knowledge points:
Radical Notation: The notation $\sqrt[n]{x}$ represents the $n$-th root of $x$. In this case, $\sqrt[5]{x}$ is the fifth root of $x$.
Properties of Exponents: $a^{mn} = (a^m)^n$ and $(ab)^n = a^n b^n$. These properties allow us to rewrite expressions to simplify them.
Simplifying Radicals: When an expression under a radical has a power that is a multiple of the index of the radical, we can pull out terms. For example, $\sqrt[n]{a^{kn}} = a^k$.
Factoring Exponents: We can factor exponents to rewrite them in a form that is easier to simplify. For example, $a^{12}$ can be factored into $a^{10} \cdot a^2$.
Combining Like Terms: When simplifying expressions, we can combine like terms to make the expression simpler. In the context of radicals, this might involve combining terms under the radical that have the same base and exponent.
By applying these principles, we can simplify the given expression by rewriting and extracting terms from under the radical, resulting in a simplified form.
