Simplify The Following Fraction In Simplest Form Using Only Positive Exponents:$\frac{10 D^9}{\left(2 D^2\right)^3}$

Feb 27, 2025 by ADMIN 117 views

**Simplifying Fractions with Exponents: A Step-by-Step Guide**

Introduction

Fractions with exponents can be a challenging topic in mathematics, but with the right approach, they can be simplified with ease. In this article, we will explore the process of simplifying fractions with exponents, using the given fraction $\frac{10 d^9}{\left(2 d^2\right)^3}$ as an example.

Understanding Exponents

Before we dive into simplifying the fraction, let's take a moment to understand what exponents are. An exponent is a small number that is placed above and to the right of a base number, indicating how many times the base number should be multiplied by itself. For example, $d^2$ means $d$ multiplied by itself twice, or $d \times d$ .

Simplifying the Fraction

Now that we have a basic understanding of exponents, let's simplify the given fraction. To simplify a fraction with exponents, we need to follow these steps:

Simplify the numerator: The numerator is the top part of the fraction, and it is given as $10 d^9$ . We can simplify this by factoring out any common factors. In this case, we can factor out a $10$ and a $d^9$ .
Simplify the denominator: The denominator is the bottom part of the fraction, and it is given as $\left(2 d^2\right)^3$ . We can simplify this by using the power of a power rule, which states that $\left(a^m\right)^n = a^{m \times n}$ .
Cancel out common factors: Once we have simplified the numerator and denominator, we can cancel out any common factors.

Step 1: Simplify the Numerator

The numerator is given as $10 d^9$ . We can simplify this by factoring out a $10$ and a $d^9$ .

import sympy as sp

# Define the variables
d = sp.symbols('d')

# Simplify the numerator
numerator = 10 * d**9
simplified_numerator = sp.factor(numerator)
print(simplified_numerator)

Step 2: Simplify the Denominator

The denominator is given as $\left(2 d^2\right)^3$ . We can simplify this by using the power of a power rule.

# Simplify the denominator
denominator = (2 * d**2)**3
simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

Step 3: Cancel Out Common Factors

Once we have simplified the numerator and denominator, we can cancel out any common factors.

# Cancel out common factors
simplified_fraction = sp.cancel(simplified_numerator / simplified_denominator)
print(simplified_fraction)

Conclusion

In this article, we have simplified the fraction $\frac{10 d^9}{\left(2 d^2\right)^3}$ using only positive exponents. We have followed the steps of simplifying the numerator, simplifying the denominator, and canceling out common factors. The final simplified fraction is $\frac{5 d}{2}$ .

Final Answer

The final answer is $\boxed{\frac{5 d}{2}}$ .

Additional Resources

For more information on simplifying fractions with exponents, check out the following resources:

Khan Academy: Simplifying Exponents
Mathway: Simplifying Fractions with Exponents
Wolfram Alpha: Simplifying Fractions with Exponents

FAQs

Q: What is the difference between a positive exponent and a negative exponent? A: A positive exponent indicates how many times the base number should be multiplied by itself, while a negative exponent indicates how many times the base number should be divided by itself.
Q: How do I simplify a fraction with a negative exponent? A: To simplify a fraction with a negative exponent, you can use the rule that $a^{-m} = \frac{1}{a^m}$ .
Q: Can I simplify a fraction with a variable exponent? A: Yes, you can simplify a fraction with a variable exponent by using the rules of exponents.
Simplifying Fractions with Exponents: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the process of simplifying fractions with exponents. However, we know that there are many more questions and scenarios that can arise when working with fractions and exponents. In this article, we will address some of the most frequently asked questions and provide additional guidance on simplifying fractions with exponents.

Q&A

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates how many times the base number should be multiplied by itself, while a negative exponent indicates how many times the base number should be divided by itself.

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, you can use the rule that $a^{-m} = \frac{1}{a^m}$ . For example, $\frac{1}{d^{-3}} = d^3$ .

Q: Can I simplify a fraction with a variable exponent?

A: Yes, you can simplify a fraction with a variable exponent by using the rules of exponents. For example, $\frac{d^2}{d^3} = d^{2-3} = d^{-1} = \frac{1}{d}$ .

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, you can add the exponents. For example, $d^2 \times d^3 = d^{2+3} = d^5$ .

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, you can subtract the exponents. For example, $\frac{d^5}{d^3} = d^{5-3} = d^2$ .

Q: Can I simplify a fraction with a zero exponent?

A: Yes, you can simplify a fraction with a zero exponent by using the rule that $a^0 = 1$ . For example, $\frac{d^0}{d^3} = \frac{1}{d^3}$ .

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, you can multiply the exponents. For example, $(d^2)^3 = d^{2 \times 3} = d^6$ .

Q: Can I simplify a fraction with a fractional exponent?

A: Yes, you can simplify a fraction with a fractional exponent by using the rule that $a^{m/n} = \sqrt[n]{a^m}$ . For example, $\frac{d^{1/2}}{d^{3/2}} = \frac{\sqrt{d}}{\sqrt{d^3}} = \frac{\sqrt{d}}{d^{3/2}} = \frac{1}{d^{3/2}}$ .

Additional Tips and Tricks

When simplifying fractions with exponents, always start by simplifying the numerator and denominator separately.
Use the rules of exponents to simplify the fraction, and then cancel out any common factors.
Be careful when working with negative exponents, as they can be tricky to handle.
Practice, practice, practice! The more you practice simplifying fractions with exponents, the more comfortable you will become with the rules and procedures.

Conclusion

Simplifying fractions with exponents can be a challenging task, but with practice and patience, you can master the rules and procedures. Remember to always start by simplifying the numerator and denominator separately, and then use the rules of exponents to simplify the fraction. With these tips and tricks, you will be well on your way to becoming a pro at simplifying fractions with exponents!