Problem

Simplify square root of 50t^7w^11

The question asks to perform a simplification of the square root expression given. The expression within the square root is "50t^7w^11", and the goal is to express this as simply as possible, presumably in a form where the exponents are reduced to their simplest terms and any perfect squares are factored out of the square root to simplify the expression.

50t7w11

Answer

Expert–verified

Solution:

Step:1

Decompose 50t7w11 into (5t3w5)22tw.

Step:1.1

Extract 25 from 50 as 252t7w11.

Step:1.2

Express 25 as 52 to get 522t7w11.

Step:1.3

Separate t6 as 522(t6t)w11.

Step:1.4

Represent t6 as (t3)2 yielding 522((t3)2t)w11.

Step:1.5

Isolate w10 as 522((t3)2t)(w10w).

Step:1.6

Reformulate w10 as (w5)2 to obtain 522((t3)2t)((w5)2w).

Step:1.7

Rearrange to place t as 522((t3)2)(w5)2tw.

Step:1.8

Shift the 2 to get (52((t3)2))(w5)22tw.

Step:1.9

Rephrase (52((t3)2))(w5)2 as (5t3w5)2 resulting in (5t3w5)22tw.

Step:1.10

Enclose with parentheses to form ((5t3w5)2)2(tw).

Step:1.11

Finalize the parentheses as ((5t3w5)2)(2tw).

Step:2

Extract terms from the radical to get 5t3w52tw.

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, t6 can be written as (t3)2 because 3×2=6.

  3. Square Roots: Recognizing that the square root of a perfect square, such as 52, 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=25×2 or 50=52×2.

  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 a2=a if a is non-negative, and that ab=ab, which allows for the separation of terms under a square root.

By applying these principles, the original expression 50t7w11 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.

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