Simplify fifth root of 96t^11u^15

Solution:

Step 1: Express the given expression in a form suitable for simplification

Start by rewriting $96t^{11}{u}^{15}$ as a product of terms raised to the power of 5 and a remaining term.

Step 1.1: Factor out 32 from 96

Recognize that $96$ can be divided by $32$, which is a perfect fifth power: $\sqrt[5]{32\cdot 3t^{11}{u}^{15}}$.

Step 1.2: Represent 32 as a power of 2

Notice that $32$ is $2^5$: $\sqrt[5]{2^5\cdot 3t^{11}{u}^{15}}$.

Step 1.3: Extract $t^{10}$ from $t^{11}$

Separate $t^{11}$ into $t^{10}$ and $t$: $\sqrt[5]{2^5\cdot 3(t^{10}t)u^{15}}$.

Step 1.4: Rewrite $t^{10}$ as $(t^2{)}^5$

Express $t^{10}$ as the fifth power of $t^2$: $\sqrt[5]{2^5\cdot 3((t^2{)}^{5}t)u^{15}}$.

Step 1.5: Rewrite $u^{15}$ as $(u^3{)}^5$

Similarly, express $u^{15}$ as the fifth power of $u^3$: $\sqrt[5]{2^5\cdot 3((t^2{)}^{5}t)(u^3{)}^5}$.

Step 1.6: Rearrange the term $t$

Move the single $t$ term: $\sqrt[5]{2^5\cdot 3((t^2{)}^5)(u^3{)}^{5}t}$.

Step 1.7: Rearrange the term $3$

Place the $3$ outside the radical: $\sqrt[5]{(2^5((t^2{)}^5))(u^3{)}^5\cdot 3t}$.

Step 1.8: Combine terms under the fifth root

Combine the terms that are perfect fifth powers: $\sqrt[5]{(2t^2{u}^3{)}^5\cdot 3t}$.

Step 1.9: Add parentheses for clarity

Enclose the terms inside the radical with parentheses for better readability: $\sqrt[5]{((2t^2{u}^3{)}^5)\cdot (3t)}$.

Step 2: Simplify the expression by pulling out terms from under the radical

Extract the terms that are perfect fifth powers from under the radical: $2t^2{u}^3\sqrt[5]{3t}$.

Knowledge Notes:

1. Fifth Root: The fifth root of a number $x$ is a number $y$ such that $y^5=x$. It is denoted as $\sqrt[5]{x}$.

2. Exponent Laws: When simplifying expressions with exponents, remember that $(a^m{)}^n=a^{mn}$ and $a^m\cdot a^n=a^{m+n}$.

3. Factoring: Factoring involves expressing a number or expression as a product of its factors. For example, $96$ can be factored into $32\cdot 3$, where $32$ is a perfect fifth power, $2^5$.

4. Simplifying Radicals: When simplifying radicals, any factor raised to a power that is a multiple of the index can be taken out of the radical. For example, $\sqrt[5]{a^5}=a$.

5. Combining Like Terms: When simplifying expressions, terms with the same base and exponent can be combined. For instance, $a^m\cdot a^n$ can be combined into $a^{m+n}$ if $m$ and $n$ are integers.

6. Perfect Powers: A perfect fifth power is a number that can be expressed as $a^5$ for some integer $a$. For example, $32$ is a perfect fifth power because it is $2^5$.