Simplify (x^2+14x+49)/((x+7)^3 square root of x+7)

Solution:

Simplification Process

Step 1: Apply the perfect square trinomial rule

1. Express $49$ as $7^2$.

   $$\frac{x^2+14x+7^2}{((x+7{)}^3\sqrt{x+7}}$$

2. Confirm the middle term is twice the product of the square roots of the first and third terms.

   $$14x=2\cdot x\cdot 7$$

3. Rewrite the numerator as a perfect square trinomial.

   $$\frac{(x+7{)}^2}{((x+7{)}^3\sqrt{x+7}}$$

Step 2: Reduce the fraction by canceling common factors

1. Introduce a multiplicative identity.

   $$\frac{(x+7{)}^2\cdot 1}{(x+7{)}^3\sqrt{x+7}}$$

2. Simplify by canceling out $(x+7{)}^2$.

   $$\frac{1}{(x+7)\sqrt{x+7}}$$

Step 3: Rationalize the denominator

Multiply by the conjugate of the denominator.

$$\frac{1}{(x+7)\sqrt{x+7}}\cdot \frac{\sqrt{x+7}}{\sqrt{x+7}}$$

Step 4: Simplify the expression

1. Combine the terms in the denominator.

   $$\frac{\sqrt{x+7}}{(x+7)(\sqrt{x+7}{)}^2}$$

2. Apply exponent rules to simplify.

   $$\frac{\sqrt{x+7}}{(x+7)(x+7)}$$

Step 5: Final simplification

Combine like terms in the denominator.

$$\frac{\sqrt{x+7}}{(x+7{)}^2}$$

Knowledge Notes:

To solve the given problem, several mathematical concepts and rules are applied:

1. Perfect Square Trinomial: A polynomial of the form $a^2+2ab+b^2$ which can be factored into $(a+b{)}^2$.

2. Common Factor: A term that is present in both the numerator and denominator of a fraction, which can be canceled out.

3. Rationalizing the Denominator: The process of eliminating radicals from the denominator of a fraction by multiplying the numerator and denominator by the conjugate of the denominator.

4. Exponent Rules: Mathematical rules that apply to expressions with exponents, such as the power rule ($a^m\cdot a^n=a^{m+n}$) and the rule for raising a power to a power ($(a^m{)}^n=a^{mn}$).

5. Conjugate: In the context of rationalizing, the conjugate of $\sqrt{a}$ is $\sqrt{a}$ itself, and when multiplied, it results in $a$ because $\sqrt{a}\cdot \sqrt{a}=a$.

By applying these concepts, we can simplify the given expression to its simplest form.