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Chapter 10: Problem 24

Simplify by taking the roots of the numerator and the denominator. Assume that all variables represent positive numbers. $$ \sqrt[5]{\frac{243 a^{9}}{b^{13}}} $$

Short Answer

Expert verified
\(\frac{3 a^{9/5}}{b^{13/5}} \)

Step by step solution

01

Rewrite the Radical Expression

Rewrite the given expression \(\frac{243 a^{9}}{b^{13}}\) under a single radical as \(\frac{243 a^{9}}{b^{13}}\).
02

Apply the Fifth Root to Both Numerator and Denominator

Apply the fifth root to the entire expression: \(\frac{\root 5 \big(243 a^{9}\big)}{\root 5 \big(b^{13}\big)} \).
03

Simplify the Fifth Root of the Numerator

Simplify \(\root 5 \big(243 a^{9}\big) \) by finding the fifth root of 243 and using the properties of exponents: \(\root 5 \big(243\big) = 3\) and \(\root 5 \big(a^{9}\big) = a^{9/5} \). Thus, \(\root 5 \big(243 a^{9}\big) = 3 a^{9/5} \).
04

Simplify the Fifth Root of the Denominator

Simplify \(\root 5 \big(b^{13}\big) \) using the properties of exponents: \(\root 5 \big(b^{13}\big) = b^{13/5} \).
05

Combine Simplified Parts

Combine the simplified numerator and denominator: \(\frac{3 a^{9/5}}{b^{13/5}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

fifth roots
The concept of **fifth roots** involves finding a number that, when multiplied by itself five times, results in the original number. For instance, the fifth root of 243 is 3 because \(3 \times 3 \times 3 \times 3 \times 3 = 243\).
To break it down:
  • The fifth root is often written as \( \root 5 \quad ( ... ) \).
  • You can apply this operation to both numbers and variable expressions, like \( a^9 \) or \( b^{13} \).
When working with variables and exponents, the exponent inside the root is divided by 5. For example, \( \root 5 \big(a^{9}\big) = a^{9/5} \).
Applying the fifth root allows simplification of more complex expressions by breaking them down into more manageable parts.

**Example**:
  • Fifth root of the numerator: \( \root 5 \big(243 a^{9}\big) = 3 a^{9/5} \)
  • Fifth root of the denominator: \( \root 5 \big(b^{13}\big) = b^{13/5} \)
These simplified elements can then be combined to form the final simplified expression.
properties of exponents
Understanding the **properties of exponents** is crucial for simplifying radical expressions. Here are some important properties:
  • **Product of Powers**: \(a^m \times a^n = a^{m+n} \)
  • **Quotient of Powers**: \(a^m \big/ a^n = a^{m-n} \)
  • **Power of a Power**: \( (a^m)^n = a^{m \times n} \)
When a root is involved, you divide the exponents by the root:
For example: \( \root 5 \big(a^9\big) = a^{9/5} \).
This property makes it easier to handle large numbers and complex variables.
In our exercise:
  • We applied the fifth root to the numerator: \( \root 5 \big(a^9\big) = a^{9/5} \)
  • We applied the fifth root to the denominator: \( \root 5 \big(b^{13}\big) = b^{13/5} \)
  • Knowing these properties helps simplify each part of the expression efficiently.
simplification of rational expressions
**Simplification of rational expressions** involves reducing the numerator and the denominator to their simplest forms.
This often includes:
  • Finding common factors and cancelling them out.
  • Using the properties of exponents to reduce complex parts.
  • Applying roots and radicals effectively.
In our example, the rational expression \( \frac{243a^9}{b^{13}} \) was simplified using fifth roots.
Step-by-step:
  • Rewrite the expression under a single radical: \( \frac{\root 5 \big(243a^9\big)}{\root 5 \big(b^{13}\big)} \)
  • Apply the fifth root to both parts: \(\frac{3a^{9/5}}{b^{13/5}} \)
  • Combine the results to get: \( \frac{3a^{9/5}}{b^{13/5}} \)
By understanding each step deeply, you're able to simplify effectively and correctly.
Practicing these problems enhances your ability to handle more complicated rational expressions with ease.

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Most popular questions from this chapter

The absolute value of a complex number \(a+b i\) is its distance from the origin. Using the distance formula, we have \(|a+b i|=\sqrt{a^{2}+b^{2}} .\) Find the absolute value of each complex number. $$ |3+4 i| $$

Multiply and simplify. Assume that no radicands were formed by raising negative numbers to even powers. $$ \sqrt{10} \sqrt{14} $$

f(x)\( and \)g(x)\( are as given. Find \)(f \cdot g)(x) \cdot$ Assume that all variables represent non negativereal numbers. $$f(x)=\sqrt[4]{x}, g(x)=2 \sqrt{x}-\sqrt[3]{x^{2}}$$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[4]{x^{2} y^{3}}}{\sqrt[3]{x y}}$$

Let \(f(x)=x^{2} .\) Find each of the following. $$f(5+\sqrt{2})$$
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