Simplify sixth root of 96x^18y^24
The provided mathematical expression involves simplifying a radical expression with an index of six, known as the sixth root. The task is to simplify this expression fully, reducing it to its simplest form if possible. This simplification process would likely involve factoring the radicand, which is the number inside the radical sign, in this case, 96x^18y^24. The goal is to express the radicand as products of powers that are multiples of the root's index—in this case, multiples of six—so that they can be taken out of the radical as whole numbers or polynomial expressions.
$\sqrt[6]{96 x^{18} y^{24}}$
Answer
Solution:
Step 1: Express the given expression in a form that facilitates simplification.
Rewrite the expression $96x^{18}y^{24}$ as a product of a perfect sixth power and the remaining factor, $96$.
Step 1.1: Factor $x^{18}$ as a sixth power.
Recognize that $x^{18}$ can be expressed as $(x^3)^6$. Thus, we have $\sqrt[6]{96(x^3)^6y^{24}}$.
Step 1.2: Factor $y^{24}$ as a sixth power.
Similarly, $y^{24}$ can be expressed as $(y^4)^6$. Now the expression is $\sqrt[6]{96(x^3)^6(y^4)^6}$.
Step 1.3: Rearrange the terms for clarity.
Position the constant $96$ at the end of the expression to emphasize the perfect sixth powers: $\sqrt[6]{(x^3)^6(y^4)^6 \cdot 96}$.
Step 1.4: Combine the sixth powers under a single radical.
Combine the terms with exponents to form a single sixth power: $\sqrt[6]{(x^3y^4)^6 \cdot 96}$.
Step 2: Simplify the sixth root.
Extract the sixth power from under the radical, leaving the constant inside: $x^3y^4\sqrt[6]{96}$.
Knowledge Notes:
To simplify a radical expression, especially one involving roots other than square roots, it's helpful to express the terms under the radical as powers that match the index of the root. In this case, we are dealing with a sixth root, so we want to express the variables as sixth powers if possible.
Exponent Laws: When simplifying expressions with exponents, remember that $(a^m)^n = a^{mn}$. This is why $x^{18}$ can be written as $(x^3)^6$ and $y^{24}$ as $(y^4)^6$.
Radicals and Exponents: The expression $\sqrt[n]{a^n}$ is equivalent to $a$ when $n$ is odd and $|a|$ when $n$ is even, provided $a$ is real. This is because the nth root and the nth power are inverse operations.
Simplifying Radicals: When simplifying radicals, any factor under the radical that is a perfect power of the index can be taken out of the radical. In this problem, we have sixth powers under a sixth root, so they can be taken out directly.
Multiplication under Radicals: When factors are multiplied under a radical, they can be combined into a single term or separated if it helps in simplification. In this problem, we combine the sixth powers before taking them out of the radical.
By applying these principles, we can simplify the given radical expression to its simplest form.
