Simplify square root of 28x^9

The question is asking for the simplification of a mathematical expression, which in this case is the square root of a monomial (a single-term algebraic expression). The monomial provided is "28x^9". To simplify the square root of a monomial, you typically need to find the square root of the coefficient (the numerical part, which is 28 in this instance) and the square root of the variable part raised to a power (which is x to the power of 9). The simplification process involves breaking down the monomial into factors that are perfect squares and simplifying them inside and outside the square root respectively. For the variables with exponents, this involves reducing the exponents while following the rules of exponents for square roots.

$\sqrt{28 x^{9}}$

Answer

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Solution:

Step 1: Decompose the expression

Break down $28 x^{9}$ into prime factors and perfect squares.

Step 1.1

Extract the factor of $4$ from $28$ to get $\sqrt{4 \cdot 7 x^{9}}$ .

Step 1.2

Express $4$ as $2^{2}$ to obtain $\sqrt{2^{2} \cdot 7 x^{9}}$ .

Step 1.3

Separate out $x^{8}$ from $x^{9}$ to rewrite as $\sqrt{2^{2} \cdot 7 (x^{8} x)}$ .

Step 1.4

Represent $x^{8}$ as $(x^{4})^{2}$ to get $\sqrt{2^{2} \cdot 7 ((x^{4})^{2} x)}$ .

Step 1.5

Rearrange to place $7$ appropriately: $\sqrt{2^{2} (x^{4})^{2} \cdot 7 x}$ .

Step 1.6

Combine $2^{2}$ and $(x^{4})^{2}$ to form $((2 x^{4})^{2})$ , resulting in $\sqrt{((2 x^{4})^{2}) \cdot 7 x}$ .

Step 1.7

Enclose the expression in parentheses: $\sqrt{((2 x^{4})^{2}) \cdot (7 x)}$ .

Step 2: Simplify the radical

Extract the square root of the perfect square and simplify to $2 x^{4} \sqrt{7 x}$ .

Knowledge Notes:

To simplify a square root involving variables and coefficients, we follow these steps:

Prime Factorization: Break down the coefficient into its prime factors to identify any perfect squares.
Perfect Square Identification: Recognize that any variable with an even exponent can be expressed as a perfect square. For example, $x^{8} = (x^{4})^{2}$ .
Extraction of Perfect Squares: Since the square root of a perfect square is an integer, we can pull out the square root of any perfect square factors from under the radical.
Simplification: After extracting perfect squares, simplify the expression by multiplying the factors outside the radical and keeping the remaining factors within the radical.

In this problem, we have used these principles to simplify $\sqrt{28 x^{9}}$ . We identified $4$ as a perfect square factor of $28$ and $x^{8}$ as a perfect square of $x^{9}$ . We then extracted these to get $2 x^{4}$ outside the square root, leaving us with $\sqrt{7 x}$ inside the square root. The final simplified form is $2 x^{4} \sqrt{7 x}$ .