Simplify 33 root of 1/(u^-3*u^-8)
In this problem, you are being asked to simplify a mathematical expression which involves the cube root (denoted as "33 root" or more commonly as the cube root symbol ∛) of a fraction with variables in the denominator raised to negative exponents. The fraction is 1 divided by \( u^{-3} \cdot u^{-8} \), and your task is to simplify this expression, most likely using the properties of exponents and roots to arrive at a simpler expression that no longer has negative exponents or a fraction under the cube root.
$\sqrt[33]{\frac{1}{u^{- 3} \cdot u^{- 8}}}$
Answer
Solution:
Simplification Process
Step:1
Commence by simplifying the expression through the elimination of common factors.
Step:1.1
Apply the rule for negative exponents to transfer $u^{-3}$ to the numerator: $\frac{1}{b^{-n}} = b^n$. The expression becomes $\sqrt[33]{\frac{u^3}{u^{-8}}}$.
Step:1.2
Similarly, apply the negative exponent rule to move $u^{-8}$ to the numerator: $\frac{1}{b^{-n}} = b^n$. The expression now is $\sqrt[33]{u^3 \cdot u^8}$.
Step:1.3
Combine $u^3$ and $u^8$ by summing their exponents.
Step:1.3.1
Utilize the exponent multiplication rule $a^m \cdot a^n = a^{m+n}$ to merge the exponents into a single term: $\sqrt[33]{u^{3+8}}$.
Step:1.3.2
Perform the addition of the exponents $3$ and $8$: $\sqrt[33]{u^{11}}$.
Step:2
Transform $\sqrt[33]{u^{11}}$ into a nested radical: $\sqrt[3]{\sqrt[11]{u^{11}}}$.
Step:3
Extract terms from under the radical, presuming all numbers involved are real: $\sqrt[3]{u}$.
Knowledge Notes:
To solve the given problem, we employ several rules of exponents and radicals:
Negative Exponent Rule: For any nonzero number $b$ and integer $n$, $b^{-n} = \frac{1}{b^n}$. This allows us to move a term with a negative exponent from the denominator to the numerator or vice versa by changing the sign of the exponent.
Multiplication of Powers with the Same Base: When multiplying powers with the same base, we add the exponents, as per the rule $a^m \cdot a^n = a^{m+n}$.
Radicals and Rational Exponents: A radical can also be expressed as a number with a rational exponent. The $n$th root of $a$ is equivalent to $a^{1/n}$, and for nested radicals, we can use the property $\sqrt[n]{\sqrt[m]{a}} = a^{\frac{1}{n} \cdot \frac{1}{m}}$.
Simplifying Radicals: When simplifying radicals, if the exponent of the term inside the radical is equal to or greater than the index of the radical, we can simplify by taking the term out of the radical. In this case, since we have a 33rd root, we cannot directly simplify $\sqrt[33]{u^{11}}$ further without additional context or constraints.
By applying these rules systematically, we can simplify the given expression to its most reduced form.
