Problem 61
Question
For the following problems, simplify each expressions. $$ \frac{2}{1-\sqrt{a}} $$
Step-by-Step Solution
Verified Answer
Question: Simplify the expression, $$\frac{2}{1 - \sqrt{a}}$$.
Answer: The simplified expression is $$\frac{2 + 2\sqrt{a}}{1 - a}$$.
1Step 1: Identify the conjugate of the denominator
The conjugate of the denominator (1 - √a) is 1 + √a.
2Step 2: Multiply the expression by the conjugate
Multiply both the numerator and the denominator of the fraction by the conjugate (1 + √a) to simplify the expression without changing its value. This will give us:
$$
\frac{2}{1-\sqrt{a}} \cdot \frac{1 + \sqrt{a}}{1 + \sqrt{a}}
$$
3Step 3: Simplify the numerator
Multiply 2 by (1 + √a) in the numerator:
$$
2(1 + \sqrt{a}) = 2 + 2\sqrt{a}
$$
4Step 4: Simplify the denominator
To simplify the denominator, multiply (1 - √a) by (1 + √a) using the difference of squares formula. This formula states that for any two terms a and b, (a - b)(a + b) = a^2 - b^2. In our case, a = 1 and b = √a:
$$
(1 - \sqrt{a})(1 + \sqrt{a}) = 1^2 - (\sqrt{a})^2 = 1 - a
$$
5Step 5: Write the simplified expression
Now that we have simplified the numerator and the denominator, we can write the simplified expression:
$$
\frac{2 + 2\sqrt{a}}{1 - a}
$$
Key Concepts
Conjugate of a DenominatorDifference of SquaresRationalizing the DenominatorAlgebraic Simplification
Conjugate of a Denominator
The conjugate of a complex number involves changing the sign between two terms in a binomial. For example, if we have a denominator with a square root—like in our exercise with \(1-\sqrt{a}\)—its conjugate is simply \(1+\sqrt{a}\). Why use the conjugate? It's instrumental when we want to rationalize the denominator, a vital process in algebra.
When you multiply a radical expression by its conjugate, we eliminate the radical from the denominator. This happens because of a property known as the difference of squares, which we'll explore next. The conjugate is not just a tool for simplifying radical expressions; it's also useful in complex numbers, allowing us to find multiplicative inverses and simplifications in different math contexts.
When you multiply a radical expression by its conjugate, we eliminate the radical from the denominator. This happens because of a property known as the difference of squares, which we'll explore next. The conjugate is not just a tool for simplifying radical expressions; it's also useful in complex numbers, allowing us to find multiplicative inverses and simplifications in different math contexts.
Difference of Squares
The difference of squares is a pattern in algebra that shows up frequently in various mathematical contexts. It refers to the identity \((a-b)(a+b) = a^2 - b^2\). It's a quick way to expand expressions or to factor them. In the context of simplifying radical expressions, we use the difference of squares to eliminate radicals when we multiply a binomial by its conjugate.
As seen in the exercise solution, \(1 - \sqrt{a}\) multiplied by \(1 + \sqrt{a}\) becomes \((1)^2 - (\sqrt{a})^2\), which simplifies to \((1 - a\)), a radical-free expression. Grasping this concept helps with factoring, solving equations, and rationalizing denominators—three critical areas in algebra that often intersect.
As seen in the exercise solution, \(1 - \sqrt{a}\) multiplied by \(1 + \sqrt{a}\) becomes \((1)^2 - (\sqrt{a})^2\), which simplifies to \((1 - a\)), a radical-free expression. Grasping this concept helps with factoring, solving equations, and rationalizing denominators—three critical areas in algebra that often intersect.
Rationalizing the Denominator
The process of rationalizing the denominator removes any radicals from the denominator of a fraction. This step makes the expression cleaner and easier to work with, especially for further algebraic operations or calculus limits.
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. By doing so, we leverage the difference of squares to our advantage and ensure that the radical is eliminated from the denominator, as it's a standard convention for a proper mathematical expression. In essence, this doesn't change the value of the expression—since we're essentially multiplying by one—but it does change its form into something more acceptable for mathematicians.
To rationalize the denominator, we multiply both the numerator and denominator by the conjugate of the denominator. By doing so, we leverage the difference of squares to our advantage and ensure that the radical is eliminated from the denominator, as it's a standard convention for a proper mathematical expression. In essence, this doesn't change the value of the expression—since we're essentially multiplying by one—but it does change its form into something more acceptable for mathematicians.
Algebraic Simplification
Lastly, algebraic simplification is a broad term that involves reducing expressions into their simplest form. It could involve factoring, expanding, combining like terms, or even rationalizing the denominator. The goal of simplification is to make the expression easier to understand and use in subsequent calculations.
Our exercise showcases an instance where we simplify a fraction with a radical in the denominator. It's often one of the initial steps in algebra problems and serves as the foundation for more complicated operations. Mastering algebraic simplification is critical for students since it's a frequent prerequisite for higher-level math problems involving calculus, functions, and problem-solving in physics and engineering scenarios.
Our exercise showcases an instance where we simplify a fraction with a radical in the denominator. It's often one of the initial steps in algebra problems and serves as the foundation for more complicated operations. Mastering algebraic simplification is critical for students since it's a frequent prerequisite for higher-level math problems involving calculus, functions, and problem-solving in physics and engineering scenarios.
Other exercises in this chapter
Problem 61
For the following problems, simplify each of the radical expressions. $$ \sqrt{\frac{2}{7}} $$
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For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number. $$ \sqrt{2 k-1} $
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Simplify each expression by performing the indicated operation. $$ (3+\sqrt{3})(3-\sqrt{3}) $$
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