Simplify the Radical Expression ( square root of a+1-2)/( square root of a+1+2)
The question asks for the simplification of a given radical expression. Specifically, the expression provided is in the form of a fraction where the numerator is the square root of (a + 1) minus 2, and the denominator is the square root of (a + 1) plus 2. The task is to perform algebraic manipulations to simplify this expression to a more elementary or reduced form. This often involves rationalizing the denominator or applying algebraic identities to simplify the radical terms.
$\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} + 2}$
Answer
Solution:
Step:1
Rationalize the expression by multiplying the numerator and denominator by the conjugate of the denominator: $\frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} + 2} \cdot \frac{\sqrt{a + 1} - 2}{\sqrt{a + 1} - 2}$.
Step:2
Apply the conjugate multiplication to both the numerator and denominator: $\frac{(\sqrt{a + 1} - 2)(\sqrt{a + 1} - 2)}{(\sqrt{a + 1} + 2)(\sqrt{a + 1} - 2)}$.
Step:3
Use the difference of squares formula to expand the denominator: $\frac{(\sqrt{a + 1} - 2)(\sqrt{a + 1} - 2)}{(\sqrt{a + 1})^2 - (2)^2}$.
Step:4
Simplify the denominator by performing the subtraction: $\frac{(\sqrt{a + 1} - 2)(\sqrt{a + 1} - 2)}{a + 1 - 4}$.
Step:5
Simplify the numerator.
Step:5.1
Square the binomial in the numerator: $\frac{(\sqrt{a + 1} - 2)^1(\sqrt{a + 1} - 2)}{a - 3}$.
Step:5.2
Continue squaring the binomial: $\frac{(\sqrt{a + 1} - 2)^1(\sqrt{a + 1} - 2)^1}{a - 3}$.
Step:5.3
Combine the exponents using the power rule $(a^m)(a^n) = a^{m+n}$: $\frac{(\sqrt{a + 1} - 2)^{1+1}}{a - 3}$.
Step:5.4
Add the exponents: $\frac{(\sqrt{a + 1} - 2)^2}{a - 3}$.
Knowledge Notes:
Rationalizing the Denominator: This is a technique used to eliminate radicals from the denominator of a fraction. It involves multiplying the numerator and the denominator by the conjugate of the denominator.
Conjugate: The conjugate of a binomial expression is obtained by changing the sign between two terms. For example, the conjugate of $\sqrt{a + 1} + 2$ is $\sqrt{a + 1} - 2$.
Difference of Squares: This is a pattern used in algebra where the product of two conjugate binomials equals the difference of the squares of each term, i.e., $(a + b)(a - b) = a^2 - b^2$.
FOIL Method: A technique for expanding the product of two binomials, which stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.
Power Rule: In algebra, the power rule for exponents states that when multiplying two powers that have the same base, you can add the exponents, i.e., $a^m \cdot a^n = a^{m+n}$.
Squaring a Binomial: When you square a binomial, you multiply the binomial by itself, which can be done by using the FOIL method or by recognizing patterns in special products.
