Simplify the Radical Expression ( square root of a+1-2)/( square root of a+1+2)

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:

1. 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.

2. 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 $\sqrt{a+1}-2$.

3. 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., $(a+b)(a-b)=a^2-b^2$.

4. 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.

5. 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., $a^m\cdot a^n=a^{m+n}$.

6. 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.