Simplify square root of 50t^7w^11

Solution:

Step:1

Decompose $50t^7w^{11}$ into $\left(5t^3w^5\right)^2 \cdot 2tw$.

Step:1.1

Extract $25$ from $50$ as $\sqrt{25 \cdot 2t^7w^{11}}$.

Step:1.2

Express $25$ as $5^2$ to get $\sqrt{5^2 \cdot 2t^7w^{11}}$.

Step:1.3

Separate $t^6$ as $\sqrt{5^2 \cdot 2(t^6t)w^{11}}$.

Step:1.4

Represent $t^6$ as $(t^3)^2$ yielding $\sqrt{5^2 \cdot 2((t^3)^2t)w^{11}}$.

Step:1.5

Isolate $w^{10}$ as $\sqrt{5^2 \cdot 2((t^3)^2t)(w^{10}w)}$.

Step:1.6

Reformulate $w^{10}$ as $(w^5)^2$ to obtain $\sqrt{5^2 \cdot 2((t^3)^2t)((w^5)^2w)}$.

Step:1.7

Rearrange to place $t$ as $\sqrt{5^2 \cdot 2((t^3)^2)(w^5)^2tw}$.

Step:1.8

Shift the $2$ to get $\sqrt{(5^2((t^3)^2))(w^5)^2 \cdot 2tw}$.

Step:1.9

Rephrase $(5^2((t^3)^2))(w^5)^2$ as $(5t^3w^5)^2$ resulting in $\sqrt{(5t^3w^5)^2 \cdot 2tw}$.

Step:1.10

Enclose with parentheses to form $\sqrt{((5t^3w^5)^2) \cdot 2(tw)}$.

Step:1.11

Finalize the parentheses as $\sqrt{((5t^3w^5)^2) \cdot (2tw)}$.

Step:2

Extract terms from the radical to get $5t^3w^5\sqrt{2tw}$.

Knowledge Notes:

The problem involves simplifying a radical expression with variables and exponents. The key knowledge points to understand this problem are:

1. Radical Simplification: The process of simplifying square roots (or other radicals) by factoring out perfect squares and reducing the expression to simplest form.

2. Exponent Rules: Understanding how to manipulate exponents, particularly when factoring expressions. For example, $t^6$ can be written as $(t^3)^2$ because $3 \times 2 = 6$.

3. Square Roots: Recognizing that the square root of a perfect square, such as $5^2$, is simply the base of the exponent, which is $5$ in this case.

4. Factoring Numbers: Breaking down numbers into their prime factors, such as $50 = 25 \times 2$ or $50 = 5^2 \times 2$.

5. Combining Like Terms: When simplifying expressions, like terms can be combined or factored out to simplify the expression further.

6. Radical Properties: Knowing that $\sqrt{a^2} = a$ if $a$ is non-negative, and that $\sqrt{ab} = \sqrt{a}\sqrt{b}$, which allows for the separation of terms under a square root.

By applying these principles, the original expression $\sqrt{50t^7w^{11}}$ is broken down into its prime factors and perfect squares, which are then taken out of the square root to simplify the expression. The final result is a simplified radical expression with some variables still under the radical and others factored out.