Simplify The Expression: $\frac{15 X^2 Y^{10} Z^7}{5 X^3 Y^4 Z^{10}}$Write Your Answer Without Negative Exponents.

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and techniques involved in simplifying expressions. In this article, we will focus on simplifying the given expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents.

Understanding the Rules of Exponents

Before we dive into simplifying the expression, it is essential to understand the rules of exponents. The rules of exponents state that when we multiply two or more numbers with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$. Similarly, when we divide two or more numbers with the same base, we subtract their exponents. For example, $\frac{x^2}{x^3} = x^{2-3} = x^{-1}$.

Simplifying the Expression

To simplify the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$, we need to apply the rules of exponents. We can start by simplifying the coefficients. The coefficient of the numerator is 15, and the coefficient of the denominator is 5. We can simplify the coefficients by dividing them: $\frac{15}{5} = 3$.

Simplifying the Variables

Next, we need to simplify the variables. We can start by simplifying the exponents of the variables. The exponent of $x$ in the numerator is 2, and the exponent of $x$ in the denominator is 3. We can simplify the exponents by subtracting them: $x^{2-3} = x^{-1}$. Similarly, the exponent of $y$ in the numerator is 10, and the exponent of $y$ in the denominator is 4. We can simplify the exponents by subtracting them: $y^{10-4} = y^6$. Finally, the exponent of $z$ in the numerator is 7, and the exponent of $z$ in the denominator is 10. We can simplify the exponents by subtracting them: $z^{7-10} = z^{-3}$.

Writing the Answer without Negative Exponents

Now that we have simplified the expression, we need to write the answer without negative exponents. To do this, we can use the rule that states that $x^{-n} = \frac{1}{x^n}$. Applying this rule to the expression, we get: $\frac{3 x^{-1} y^6 z^{-3}}{1} = \frac{1}{3 x y^6 z^3}$.

Conclusion

In conclusion, simplifying the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents requires applying the rules of exponents and using the rule that states that $x^{-n} = \frac{1}{x^n}$. By following these steps, we can simplify the expression and write the answer without negative exponents.

Frequently Asked Questions

• Q: What is the rule for simplifying exponents? A: The rule for simplifying exponents states that when we multiply two or more numbers with the same base, we add their exponents. When we divide two or more numbers with the same base, we subtract their exponents.
• Q: How do we simplify negative exponents? A: To simplify negative exponents, we can use the rule that states that $x^{-n} = \frac{1}{x^n}$.
• Q: What is the final answer to the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents? A: The final answer to the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents is $\frac{1}{3 x y^6 z^3}$.

Step-by-Step Solution

1. Simplify the coefficients: $\frac{15}{5} = 3$
2. Simplify the exponents of the variables:

• $x^{2-3} = x^{-1}$
• $y^{10-4} = y^6$
• $z^{7-10} = z^{-3}$

3. Write the answer without negative exponents: $\frac{1}{3 x y^6 z^3}$

Example Problems

• Simplify the expression $\frac{12 x^4 y^2 z^5}{4 x^2 y^3 z^2}$.
• Simplify the expression $\frac{20 x^3 y^5 z^2}{5 x^2 y^2 z^4}$.
• Simplify the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$.

Practice Problems

• Simplify the expression $\frac{18 x^5 y^3 z^2}{6 x^2 y^4 z^5}$.
• Simplify the expression $\frac{25 x^4 y^6 z^3}{5 x^2 y^2 z^4}$.
• Simplify the expression $\frac{30 x^3 y^5 z^2}{10 x^2 y^3 z^5}$.

Final Answer

The final answer to the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents is $\frac{1}{3 x y^6 z^3}$.

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and techniques involved in simplifying expressions. In this article, we will focus on answering frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the rule for simplifying exponents?

A: The rule for simplifying exponents states that when we multiply two or more numbers with the same base, we add their exponents. When we divide two or more numbers with the same base, we subtract their exponents.

Q: How do we simplify negative exponents?

A: To simplify negative exponents, we can use the rule that states that $x^{-n} = \frac{1}{x^n}$.

Q: What is the difference between simplifying exponents and simplifying fractions?

A: Simplifying exponents involves simplifying the exponents of the variables in an expression, while simplifying fractions involves simplifying the coefficients and variables in a fraction.

Q: How do we simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, we can use the rules of exponents and simplify each variable separately.

Q: What is the final answer to the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents?

A: The final answer to the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents is $\frac{1}{3 x y^6 z^3}$.

Q: Can we simplify expressions with variables in the denominator?

A: Yes, we can simplify expressions with variables in the denominator by using the rule that states that $x^{-n} = \frac{1}{x^n}$.

Q: How do we simplify expressions with multiple fractions?

A: To simplify expressions with multiple fractions, we can use the rules of exponents and simplify each fraction separately.

Q: What is the difference between simplifying expressions and solving equations?

A: Simplifying expressions involves simplifying the expression itself, while solving equations involves finding the value of the variable that makes the equation true.

Q: Can we simplify expressions with variables in the numerator and denominator?

A: Yes, we can simplify expressions with variables in the numerator and denominator by using the rules of exponents and simplifying each variable separately.

Example Problems

• Simplify the expression $\frac{12 x^4 y^2 z^5}{4 x^2 y^3 z^2}$.
• Simplify the expression $\frac{20 x^3 y^5 z^2}{5 x^2 y^2 z^4}$.
• Simplify the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$.

Practice Problems

• Simplify the expression $\frac{18 x^5 y^3 z^2}{6 x^2 y^4 z^5}$.
• Simplify the expression $\frac{25 x^4 y^6 z^3}{5 x^2 y^2 z^4}$.
• Simplify the expression $\frac{30 x^3 y^5 z^2}{10 x^2 y^3 z^5}$.

Final Answer

The final answer to the expression $\frac{15 x^2 y^{10} z^7}{5 x^3 y^4 z^{10}}$ without negative exponents is $\frac{1}{3 x y^6 z^3}$.

Additional Resources

• For more information on simplifying algebraic expressions, please refer to the following resources:
• Khan Academy: Simplifying Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Simplifying Algebraic Expressions

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it is essential to understand the rules and techniques involved in simplifying expressions. By following the rules of exponents and simplifying each variable separately, we can simplify expressions with multiple variables and variables in the numerator and denominator.