Simplify:$\left(\frac{2 A^5}{b^7 C^9}\right)^3$Entry Tips:1. Use ^ (shift + 6) For Exponents. To Type $x^2$, Type x^{\wedge}2.2. If There Is More Than One thing In The Numerator Or Denominator, Use Parentheses: (2a)/(c+4).

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Introduction

In this article, we will delve into the world of algebra and simplify the given expression $\left(\frac{2 a^5}{b^7 c^9}\right)^3$. This involves applying the rules of exponents and simplifying the resulting expression. We will break down the process step by step, making it easy to follow and understand.

Understanding Exponents

Before we begin, let's review the basics of exponents. An exponent is a small number that is placed above and to the right of a number or a variable. It represents the number of times the base is multiplied by itself. For example, $a^3$ means $a$ multiplied by itself three times, or $a \times a \times a$.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the given expression $\left(\frac{2 a^5}{b^7 c^9}\right)^3$. To do this, we need to apply the rule of exponents that states $(a^m)^n = a^{m \times n}$.

Using this rule, we can rewrite the expression as:

$$\left(\frac{2 a^5}{b^7 c^9}\right)^3 = \frac{2^3 \times (a^5)^3}{(b^7)^3 \times (c^9)^3}$$

Applying the Exponent Rule

Now that we have rewritten the expression, let's apply the exponent rule to simplify it further. We can start by simplifying the numerator and denominator separately.

In the numerator, we have $2^3 \times (a^5)^3$. Using the exponent rule, we can rewrite this as:

$$2^3 \times (a^5)^3 = 2^3 \times a^{5 \times 3} = 2^3 \times a^{15}$$

In the denominator, we have $(b^7)^3 \times (c^9)^3$. Using the exponent rule, we can rewrite this as:

$$(b^7)^3 \times (c^9)^3 = b^{7 \times 3} \times c^{9 \times 3} = b^{21} \times c^{27}$$

Simplifying the Expression Further

Now that we have simplified the numerator and denominator, let's put them together to get the final simplified expression.

$$\frac{2^3 \times a^{15}}{b^{21} \times c^{27}} = \frac{8 \times a^{15}}{b^{21} \times c^{27}}$$

Conclusion

In this article, we simplified the given expression $\left(\frac{2 a^5}{b^7 c^9}\right)^3$ using the rules of exponents. We broke down the process step by step, making it easy to follow and understand. By applying the exponent rule, we were able to simplify the expression and arrive at the final answer.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions like this:

• Use the exponent rule to rewrite the expression in a simpler form.
• Simplify the numerator and denominator separately.
• Use the rules of exponents to simplify the expression further.
• Check your work by plugging in values for the variables.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions like this:

• Forgetting to apply the exponent rule.
• Not simplifying the numerator and denominator separately.
• Not checking your work by plugging in values for the variables.

Real-World Applications

Simplifying expressions like this has many real-world applications. For example, in physics, you may need to simplify expressions involving exponents to solve problems involving motion and energy. In engineering, you may need to simplify expressions involving exponents to design and optimize systems.

Conclusion

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Q: What is the rule of exponents?

A: The rule of exponents states that $(a^m)^n = a^{m \times n}$. This means that when you have an exponent raised to another exponent, you can multiply the two exponents together.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the rule of exponents. This involves rewriting the expression in a simpler form by multiplying the exponents together.

Q: What is the difference between a base and an exponent?

A: A base is the number or variable that is being raised to a power, while an exponent is the number that is being raised to that power. For example, in the expression $a^3$, $a$ is the base and $3$ is the exponent.

Q: How do I simplify an expression with multiple bases and exponents?

A: To simplify an expression with multiple bases and exponents, you need to apply the rule of exponents to each base separately. This involves multiplying the exponents together for each base.

Q: What is the order of operations for simplifying expressions with exponents?

A: The order of operations for simplifying expressions with exponents is as follows:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents.
3. Multiply any numbers together.
4. Add or subtract any numbers together.

Q: How do I check my work when simplifying expressions with exponents?

A: To check your work when simplifying expressions with exponents, you can plug in values for the variables and see if the expression simplifies to the correct answer.

Q: What are some common mistakes to avoid when simplifying expressions with exponents?

A: Some common mistakes to avoid when simplifying expressions with exponents include:

• Forgetting to apply the rule of exponents.
• Not simplifying the numerator and denominator separately.
• Not checking your work by plugging in values for the variables.

Q: How do I apply the rule of exponents to simplify an expression with a negative exponent?

A: To apply the rule of exponents to simplify an expression with a negative exponent, you need to rewrite the expression in a simpler form by multiplying the exponents together. For example, in the expression $a^{-3}$, you can rewrite it as $\frac{1}{a^3}$.

Q: What are some real-world applications of simplifying expressions with exponents?

A: Some real-world applications of simplifying expressions with exponents include:

• Physics: Simplifying expressions involving exponents is necessary to solve problems involving motion and energy.
• Engineering: Simplifying expressions involving exponents is necessary to design and optimize systems.
• Computer Science: Simplifying expressions involving exponents is necessary to optimize algorithms and data structures.

Conclusion

In conclusion, simplifying expressions with exponents is a powerful tool that can help you solve problems in mathematics and other fields. By following the steps outlined in this article, you can simplify expressions like $\left(\frac{2 a^5}{b^7 c^9}\right)^3$ and arrive at the final answer.