Problem 88
Question
Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt[3]{\frac{11 a^{2}}{125 b^{6}}} $$
Step-by-Step Solution
Verified Answer
\( \frac{\sqrt[3]{11 a^2}}{5b^2} \)
1Step 1: Understand the Components
The given expression is the cube root of a fraction: \( \sqrt[3]{\frac{11 a^{2}}{125 b^{6}}} \). This can be broken down into \( \sqrt[3]{11 a^2} \) over \( \sqrt[3]{125 b^6} \).
2Step 2: Simplify the Denominator
We can simplify \( \sqrt[3]{125 b^6} \). Since \( 125 = 5^3 \), the cube root is 5. For \( b^6 \), the cube root is \( b^2 \) since \( b^6 = (b^2)^3 \). So, \( \sqrt[3]{125 b^6} = 5b^2 \).
3Step 3: Simplify the Numerator
For the numerator \( \sqrt[3]{11 a^2} \), since 11 and \( a^2 \) are not perfect cubes, it remains as \( \sqrt[3]{11} \times \sqrt[3]{a^2} \). This is simplified to \( \sqrt[3]{11 a^2} \).
4Step 4: Combine Simplified Parts
Put the simplified numerator and denominator together to get the final expression: \[ \frac{\sqrt[3]{11 a^2}}{5b^2} \]. This is the simplest form as \( 11 \) and \( a^2 \) are not perfect cubes.
Key Concepts
Cube RootsAlgebraic FractionsSimplifying Expressions
Cube Roots
Cube roots are a type of radical expression. They give us a number which, when multiplied by itself twice, results in the original number. For example, the cube root of 8 is 2, because when we multiply 2 by itself three times (2 × 2 × 2), we get 8. In mathematical terms, we write the cube root of a number as \( \sqrt[3]{x} \).
Understanding cube roots involves recognizing whether numbers and terms are perfect cubes. A perfect cube is simply a number that can be written as another integer raised to the power of 3. For instance, 27 is a perfect cube because it equals 3³.
When dealing with cube roots in expressions such as \( \sqrt[3]{11 a^2} \), if the numbers or terms inside are not perfect cubes, they can't be simplified further using basic cube root rules, and thus remain under the radical.
Understanding cube roots involves recognizing whether numbers and terms are perfect cubes. A perfect cube is simply a number that can be written as another integer raised to the power of 3. For instance, 27 is a perfect cube because it equals 3³.
When dealing with cube roots in expressions such as \( \sqrt[3]{11 a^2} \), if the numbers or terms inside are not perfect cubes, they can't be simplified further using basic cube root rules, and thus remain under the radical.
Algebraic Fractions
An algebraic fraction is a fraction that contains algebraic expressions in its numerator, denominator, or both. For example, \( \frac{11a^2}{125b^6} \) is an algebraic fraction because it contains terms with variables and constants split by a division line.
To simplify algebraic fractions, we often try to reduce or break them down into smaller parts. This can involve factoring expressions, canceling common factors from the numerator and the denominator, or applying radical expressions like cube roots appropriately.
In the context of cube roots, when given an expression like \( \sqrt[3]{\frac{11 a^{2}}{125 b^{6}}} \), the fraction can be split into the separate cube roots of the numerator and the denominator. Then, we simplify each side as much as possible.
To simplify algebraic fractions, we often try to reduce or break them down into smaller parts. This can involve factoring expressions, canceling common factors from the numerator and the denominator, or applying radical expressions like cube roots appropriately.
In the context of cube roots, when given an expression like \( \sqrt[3]{\frac{11 a^{2}}{125 b^{6}}} \), the fraction can be split into the separate cube roots of the numerator and the denominator. Then, we simplify each side as much as possible.
Simplifying Expressions
Simplifying radical expressions, especially those involving fractions, is about breaking them into more manageable pieces and reducing them to their simplest form. This involves identifying perfect powers and removing them from under the radical, as seen in the cube root of the denominator from the exercise, \( \sqrt[3]{125 b^6} = 5b^2 \).
When simplifying expressions containing cube roots, the key is to handle the variables and constants separately. Only reduce those numbers or variables that are perfect cubes, such as the \( 125 \) in our example. The non-perfect cube parts, like the \( 11a^2 \) in the numerator, remain unchanged if they can't be broken down further.
Finally, the aim is to reach an expression that is free from complex or unnecessary radical parts, resulting in the simplest form possible, like \( \frac{\sqrt[3]{11 a^2}}{5b^2} \). This allows mathematicians and students to work with more straightforward and easily understandable expressions.
When simplifying expressions containing cube roots, the key is to handle the variables and constants separately. Only reduce those numbers or variables that are perfect cubes, such as the \( 125 \) in our example. The non-perfect cube parts, like the \( 11a^2 \) in the numerator, remain unchanged if they can't be broken down further.
Finally, the aim is to reach an expression that is free from complex or unnecessary radical parts, resulting in the simplest form possible, like \( \frac{\sqrt[3]{11 a^2}}{5b^2} \). This allows mathematicians and students to work with more straightforward and easily understandable expressions.
Other exercises in this chapter
Problem 88
Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \frac{b^{4 / 5} b^{4 /
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