Problem 51
Question
Division of Radicals. Divide and simplify. $$4 \sqrt{x} \div \sqrt{a}$$
Step-by-Step Solution
Verified Answer
\( \frac{4 \sqrt{x \cdot a}}{a} \)
1Step 1: Rewrite the division as a fraction
Express the division of the radicals as a fraction by placing the numerator radical over the denominator radical. So we rewrite it as \( \frac{4 \sqrt{x}}{\sqrt{a}} \)
2Step 2: Rationalize the denominator
To simplify the expression, we want a rational number in the denominator. Multiply both the numerator and denominator by \( \sqrt{a} \) to get rid of the radical in the denominator: \( \frac{4 \sqrt{x}}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{4 \sqrt{x} \sqrt{a}}{a} \)
3Step 3: Simplify the numerator
Combine the radicals in the numerator by multiplying them: \( 4 \sqrt{x} \cdot \sqrt{a} = 4 \sqrt{x \cdot a} \)
4Step 4: Write the final expression
Now we place the simplified numerator over the original denominator to get the final simplified expression: \( \frac{4 \sqrt{x \cdot a}}{a} \)
Key Concepts
Rationalizing the DenominatorSimplifying Radical ExpressionsRadical Operations
Rationalizing the Denominator
When dealing with division of radicals, we often encounter expressions where a radical is in the denominator. To simplify these expressions, we use a process called rationalizing the denominator. The goal is to eliminate the radical from the denominator so that it contains only rational numbers.
For the exercise \( 4 \sqrt{x} \div \sqrt{a} \), the denominator is a radical, which is not considered simplified. To rationalize it, we multiply both the numerator and denominator by the same radical that is in the denominator. In this case, we multiply by \( \sqrt{a} \), resulting in \( \frac{4 \sqrt{x} \sqrt{a}}{a} \). Since we are multiplying by a form of one \( \frac{\sqrt{a}}{\sqrt{a}} = 1 \), the value of the expression remains unchanged. The outcome is a simplified expression with a rational denominator, which is the preferred form for most mathematical operations.
Rationalizing the denominator is a crucial step in simplifying radical expressions and ensures we have an expression that is easy to work with in subsequent calculations.
For the exercise \( 4 \sqrt{x} \div \sqrt{a} \), the denominator is a radical, which is not considered simplified. To rationalize it, we multiply both the numerator and denominator by the same radical that is in the denominator. In this case, we multiply by \( \sqrt{a} \), resulting in \( \frac{4 \sqrt{x} \sqrt{a}}{a} \). Since we are multiplying by a form of one \( \frac{\sqrt{a}}{\sqrt{a}} = 1 \), the value of the expression remains unchanged. The outcome is a simplified expression with a rational denominator, which is the preferred form for most mathematical operations.
Rationalizing the denominator is a crucial step in simplifying radical expressions and ensures we have an expression that is easy to work with in subsequent calculations.
Simplifying Radical Expressions
Simplifying radical expressions is a key aspect of working with radicals. Simplification can involve rationalizing the denominator, as previously discussed, or it can include combining radicals. When radicals involve multiplication, as in the given exercise \( 4 \sqrt{x} \cdot \sqrt{a} \), we can simplify the expression by combining the radicals under one radical sign.
This process, shown in Step 3 of the solution, involves using the property of radicals that \( \sqrt{m} \cdot \sqrt{n} = \sqrt{m \cdot n} \). For our example, we combine them to get \( 4 \sqrt{x \cdot a} \).
Simplifying radicals not only makes expressions neater and more comprehensible but also prepares them for further algebraic manipulation, such as addition, subtraction or comparison with other expressions. The ability to simplify radicals is critical for solving equations and working with expressions in algebra, trigonometry, and calculus.
This process, shown in Step 3 of the solution, involves using the property of radicals that \( \sqrt{m} \cdot \sqrt{n} = \sqrt{m \cdot n} \). For our example, we combine them to get \( 4 \sqrt{x \cdot a} \).
Simplifying radicals not only makes expressions neater and more comprehensible but also prepares them for further algebraic manipulation, such as addition, subtraction or comparison with other expressions. The ability to simplify radicals is critical for solving equations and working with expressions in algebra, trigonometry, and calculus.
Radical Operations
Operations with radicals, including division, multiplication, addition, and subtraction, follow specific rules to ensure that the radicals are manipulated correctly. In our example involving division of radicals \( 4 \sqrt{x} \div \sqrt{a} \), the process showcases how multiplication and division of radicals work hand in hand.
When we multiply radicals with the same index, such as square roots, we can multiply the radicands directly. This is seen in Step 3, where \( \sqrt{x} \cdot \sqrt{a} \) combines into \( \sqrt{x \cdot a} \). However, to divide radicals as seen in the initial expression, we usually want to avoid leaving a radical in the denominator, which brings us back to rationalizing the denominator.
The systematic approach to radical operations ensures that they are not left as an intimidating part of algebra. Familiarity with these operations allows students to tackle more complex problems with confidence and can demystify many aspects of higher mathematics.
When we multiply radicals with the same index, such as square roots, we can multiply the radicands directly. This is seen in Step 3, where \( \sqrt{x} \cdot \sqrt{a} \) combines into \( \sqrt{x \cdot a} \). However, to divide radicals as seen in the initial expression, we usually want to avoid leaving a radical in the denominator, which brings us back to rationalizing the denominator.
The systematic approach to radical operations ensures that they are not left as an intimidating part of algebra. Familiarity with these operations allows students to tackle more complex problems with confidence and can demystify many aspects of higher mathematics.
Other exercises in this chapter
Problem 48
The volume of a cube of side \(a\) is \(a^{3} .\) If we double the length of the side, the volume becomes $$ (2 a)^{3} $$ Simplify this expression.
View solution Problem 49
Division of Radicals. Divide and simplify. $$(3+\sqrt{2}) \div(2-\sqrt{2})$$
View solution Problem 52
Division of Radicals. Divide and simplify. $$10 \div \sqrt[3]{9 x^{2}}$$
View solution Problem 53
The hypotenuse in right triangle \(A B C\), shown in Fig. \(13-3,\) is given by the Pythagorean theorem. $$ c=\sqrt{a^{2}+b^{2}} $$ Write an expression for \(c\
View solution