Problem 70
Question
Simplify each expression. $$\frac{a}{a-\sqrt{c}}$$
Step-by-Step Solution
Verified Answer
The simplified expression is \( \frac{a^2 + a\sqrt{c}}{a^2 - c} \).
1Step 1: Identify the Problem
The expression given for simplification is \( \frac{a}{a-\sqrt{c}} \). Our goal is to simplify this expression. The presence of a square root in the denominator hints that we should rationalize the denominator.
2Step 2: Rationalize the Denominator
To eliminate the square root in the denominator, multiply both the numerator and denominator of the fraction by the conjugate of the denominator, \( a+\sqrt{c} \). This will result in: \[ \frac{a}{a-\sqrt{c}} \times \frac{a+\sqrt{c}}{a+\sqrt{c}} = \frac{a(a+\sqrt{c})}{(a-\sqrt{c})(a+\sqrt{c})}. \]
3Step 3: Simplify the Numerator
Multiply out the terms in the numerator: \[ a(a+\sqrt{c}) = a^2 + a\sqrt{c}. \]
4Step 4: Simplify the Denominator
Use the formula for the difference of squares to simplify the denominator: \[ (a-\sqrt{c})(a+\sqrt{c}) = a^2 - (\sqrt{c})^2 = a^2 - c. \]
5Step 5: Combine and Simplify the Fraction
Combine the simplified numerator and denominator to form the simplified expression: \[ \frac{a^2 + a\sqrt{c}}{a^2 - c}. \] There are no further simplifications possible, as the numerator and the denominator have no common factors.
Key Concepts
Simplifying ExpressionsDifference of SquaresAlgebraic Fractions
Simplifying Expressions
Simplifying algebraic expressions often involves reducing them to their most basic form. The idea is to make the expressions easier to work with or understand. When dealing with fractions, we aim to make both the numerator and denominator clear and concise. This can involve combining like terms or canceling out terms that appear in both the numerator and the denominator. The ultimate goal is to produce a version of the expression that maintains its original value but is simpler to manipulate.
In the provided exercise, the expression has a square root in the denominator. To simplify, it was important to rationalize the denominator, which means removing any irrational numbers from it. This was achieved by multiplying the expression by the conjugate of the denominator. After this process, the simplified version no longer has a square root as part of the denominator. This makes calculations easier and the expression more straightforward.
In the provided exercise, the expression has a square root in the denominator. To simplify, it was important to rationalize the denominator, which means removing any irrational numbers from it. This was achieved by multiplying the expression by the conjugate of the denominator. After this process, the simplified version no longer has a square root as part of the denominator. This makes calculations easier and the expression more straightforward.
Difference of Squares
The difference of squares is an algebraic pattern that you will encounter frequently when simplifying expressions. It states that for any two numbers, say \(a\) and \(b\), the product of their difference and their sum is equal to \(a^2 - b^2\). That is:
In our original problem, the denominator \((a - \sqrt{c})(a + \sqrt{c})\) was simplified using this technique. The expression was rewritten as \(a^2 - c\), which helped simplify the overall expression by eliminating the square root from the denominator. Recognizing and applying this pattern can often accelerate the simplification process and is a handy tool in algebra.
- \((a - b)(a + b) = a^2 - b^2\)
In our original problem, the denominator \((a - \sqrt{c})(a + \sqrt{c})\) was simplified using this technique. The expression was rewritten as \(a^2 - c\), which helped simplify the overall expression by eliminating the square root from the denominator. Recognizing and applying this pattern can often accelerate the simplification process and is a handy tool in algebra.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both are algebraic expressions. They often involve variables and can require several steps to simplify. Simplifying such fractions is an essential skill in algebra that involves combining like terms, factoring, or rationalizing denominators.
The original expression \(\frac{a}{a-\sqrt{c}}\) is an example of an algebraic fraction that needed simplification. Though it initially looked complex due to the square root, after rationalizing the denominator by multiplying by its conjugate \(a+\sqrt{c}\), the fraction became easier to handle. The approach reduced the complexity of the expression, making it more suitable for further algebraic operations.
The original expression \(\frac{a}{a-\sqrt{c}}\) is an example of an algebraic fraction that needed simplification. Though it initially looked complex due to the square root, after rationalizing the denominator by multiplying by its conjugate \(a+\sqrt{c}\), the fraction became easier to handle. The approach reduced the complexity of the expression, making it more suitable for further algebraic operations.
- Check for common factors that can be canceled out.
- Apply rationalizing techniques to remove radicals from the denominator.
- Use factoring strategies to simplify where possible.
Other exercises in this chapter
Problem 69
For the following exercises, simplify each expression. $$ \frac{\sqrt{m n^{3}}}{a^{2} \sqrt{c-^{3}}} \cdot \frac{a^{-7} n^{-2}}{\sqrt{m^{2} c^{4}}} $$
View solution Problem 70
For the following exercises, simplify each expression. $$ \frac{a}{a-\sqrt{c}} $$
View solution Problem 71
For the following exercises, simplify each expression. $$ \frac{x \sqrt{64 y}+4 \sqrt{y}}{\sqrt{128 y}} $$
View solution Problem 71
Simplify each expression. $$\frac{x \sqrt{64 y}+4 \sqrt{y}}{\sqrt{128 y}}$$
View solution