Problem 54
Question
For the following problems, simplify each expressions. $$ \frac{\sqrt{a^{2}+3 a+2}}{\sqrt{a+1}} $$
Step-by-Step Solution
Verified Answer
Answer: The simplified form of the expression is $\sqrt{a + 2}$.
1Step 1: Simplify the numerator
First, let's simplify the expression inside the square root in the numerator, which is a quadratic expression:
$$
a^2 + 3a + 2
$$
Factor the quadratic expression:
$$
(a + 2)(a + 1)
$$
Now, the expression inside the square root in the numerator becomes:
$$
\sqrt{(a + 2)(a + 1)}
$$
2Step 2: Simplify the entire expression
Now let's simplify the entire expression by dividing the numerator by the denominator:
$$
\frac{\sqrt{(a + 2)(a + 1)}}{\sqrt{a + 1}}
$$
Divide the numerator by the denominator:
$$
\sqrt{\frac{(a + 2)(a + 1)}{a + 1}}
$$
Now, we can cancel out the \((a + 1)\) term from the numerator and the denominator:
$$
\sqrt{a + 2}
$$
And now we have the simplified expression:
$$
\sqrt{a + 2}
$$
Key Concepts
Factoring Quadratic ExpressionsSimplifying Square RootsAlgebraic Fractions
Factoring Quadratic Expressions
To simplify algebraic expressions, particularly those involving exponents, factoring is a critical skill. Quadratic expressions are polynomials of the form \( ax^2 + bx + c \) and factoring them involves breaking down the expression into products of simpler expressions that, when multiplied, give back the original quadratic.
Factoring is often done by finding two numbers that add to give the coefficient, \( b \), and multiply to give the product of \( a \) and \( c \). For the quadratic expression \( a^2 + 3a + 2 \), we looked for two numbers that add up to 3 and multiply to give 2. These numbers are 2 and 1, which leads to the factored form \( (a + 2)(a + 1) \). In many exercises, factoring allows us to simplify fractions or to cancel out terms, as seen in the given solution. It's essential to first understand what numbers will successfully factor the quadratic before attempting to simplify further.
Factoring is often done by finding two numbers that add to give the coefficient, \( b \), and multiply to give the product of \( a \) and \( c \). For the quadratic expression \( a^2 + 3a + 2 \), we looked for two numbers that add up to 3 and multiply to give 2. These numbers are 2 and 1, which leads to the factored form \( (a + 2)(a + 1) \). In many exercises, factoring allows us to simplify fractions or to cancel out terms, as seen in the given solution. It's essential to first understand what numbers will successfully factor the quadratic before attempting to simplify further.
Simplifying Square Roots
Simplifying square roots involves finding the square root of a number or, in algebra, finding an equivalent expression that might make the equation easier to work with. This is often encountered in algebraic expressions where the square root is taken over a polynomial or a fraction.
When simplifying \( \sqrt{a^2 + 3a + 2} \), for instance, we first factor the polynomial to find squares within the radicand—the number under the square root symbol—that can be taken out of the square root. After factoring, we look for common terms inside and outside of the square root that can be canceled or simplified when dividing by the denominator or when simplifying the expression as a whole. Understanding the properties of square roots, such as \( \sqrt{ab} = \sqrt{a}\sqrt{b} \), and how radicals can be simplified or manipulated is key to working with these types of expressions.
When simplifying \( \sqrt{a^2 + 3a + 2} \), for instance, we first factor the polynomial to find squares within the radicand—the number under the square root symbol—that can be taken out of the square root. After factoring, we look for common terms inside and outside of the square root that can be canceled or simplified when dividing by the denominator or when simplifying the expression as a whole. Understanding the properties of square roots, such as \( \sqrt{ab} = \sqrt{a}\sqrt{b} \), and how radicals can be simplified or manipulated is key to working with these types of expressions.
Algebraic Fractions
Algebraic fractions, much like numerical fractions, consist of a numerator and a denominator that contain algebraic expressions. Simplifying algebraic fractions may involve factoring expressions, canceling common factors, or finding common denominators.
The goal is to reduce the fraction to its simplest form. In the solution we've seen, the fraction \( \frac{\sqrt{a^{2}+3a+2}}{\sqrt{a+1}} \) is simplified. This is done by factoring the numerator and then noticing that the denominator is a factor of the numerator's radicand. Once we cancel out the common \( a + 1 \), the expression simplifies to \( \sqrt{a + 2} \). Effective simplification often hinges on recognizing these types of relationships swiftly, making the ability to factor and simplify square roots fundamental in dealing with algebraic fractions.
The goal is to reduce the fraction to its simplest form. In the solution we've seen, the fraction \( \frac{\sqrt{a^{2}+3a+2}}{\sqrt{a+1}} \) is simplified. This is done by factoring the numerator and then noticing that the denominator is a factor of the numerator's radicand. Once we cancel out the common \( a + 1 \), the expression simplifies to \( \sqrt{a + 2} \). Effective simplification often hinges on recognizing these types of relationships swiftly, making the ability to factor and simplify square roots fundamental in dealing with algebraic fractions.
Other exercises in this chapter
Problem 54
For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{1}{4}} $$
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Find each of the following products. $$ \sqrt{y-3} \sqrt{(y-3)^{5}} $$
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Simplify each expression by performing the indicated operation. $$ (2-\sqrt{7})^{2} $$
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For the following problems, simplify the expressions. $$ 4 \sqrt{11}+8 \sqrt{11} $$
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