Simplify (x^2+14x+49)/((x+7)^3 square root of x+7)
The given expression is a complex rational function that involves polynomials and a radical expression in its denominator. It has been presented as an algebraic simplification problem. The question requires the simplification of a fraction, where the numerator is a trinomial in the form of \(x^2 + bx + c\), and the denominator is the cube of a binomial \((x+7)^3\) multiplied by the square root of that binomial \( \sqrt{x+7} \). The task involves factoring the polynomial, recognizing patterns such as perfect squares, and using the properties of exponents and radicals to reduce the expression to its simplest form.
$\frac{x^{2} + 14 x + 49}{\left(\left(\right. x + 7 \left.\right)\right)^{3} \sqrt{x + 7}}$
Answer
Solution:
Simplification Process
Step 1: Apply the perfect square trinomial rule
Express $49$ as $7^2$.
$$\frac{x^2 + 14x + 7^2}{((x + 7)^3 \sqrt{x + 7}}$$
Confirm the middle term is twice the product of the square roots of the first and third terms.
$$14x = 2 \cdot x \cdot 7$$
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
Introduce a multiplicative identity.
$$\frac{(x + 7)^2 \cdot 1}{(x + 7)^3 \sqrt{x + 7}}$$
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
Combine the terms in the denominator.
$$\frac{\sqrt{x + 7}}{(x + 7) (\sqrt{x + 7})^2}$$
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:
Perfect Square Trinomial: A polynomial of the form $a^2 + 2ab + b^2$ which can be factored into $(a + b)^2$.
Common Factor: A term that is present in both the numerator and denominator of a fraction, which can be canceled out.
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.
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 ($(a^m)^n = a^{mn}$).
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.
