Fully Simplify Using Only Positive Exponents:${ \frac{125 X^5 Y^4}{5 X^8 Y^6} }$

Understanding Exponents and Simplification

When dealing with algebraic expressions, simplifying them is an essential step to make them more manageable and easier to work with. In this article, we will focus on simplifying the given expression using only positive exponents. The expression we need to simplify is $\frac{125 x^5 y^4}{5 x^8 y^6}$.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $x^3$ can be read as "x to the power of 3" and is equivalent to $x \times x \times x$. Exponents can be positive, negative, or zero. In this article, we will focus on simplifying expressions with positive exponents.

Simplifying the Given Expression

To simplify the given expression, we need to apply the rules of exponents. The expression we need to simplify is $\frac{125 x^5 y^4}{5 x^8 y^6}$. We can start by simplifying the coefficients and then simplifying the exponents.

Simplifying the Coefficients

The coefficients in the expression are 125 and 5. We can simplify these coefficients by dividing them. $\frac{125}{5} = 25$. So, the expression becomes $\frac{25 x^5 y^4}{x^8 y^6}$.

Simplifying the Exponents

Now, we need to simplify the exponents. We can do this by applying the rule of subtracting the exponents when dividing like bases. The rule states that $\frac{x^m}{x^n} = x^{m-n}$. We can apply this rule to the expression.

$\frac{x^5}{x^8} = x^{5-8} = x^{-3}$

$\frac{y^4}{y^6} = y^{4-6} = y^{-2}$

So, the expression becomes $25 x^{-3} y^{-2}$.

Rewriting the Expression with Positive Exponents

We are asked to simplify the expression using only positive exponents. To do this, we need to rewrite the expression with positive exponents. We can do this by applying the rule of multiplying the exponents when raising a power to a power. The rule states that $(x^m)^n = x^{m \times n}$. We can apply this rule to the expression.

$25 x^{-3} y^{-2} = 25 (x^{-3} \times y^{-2})$

$= 25 (x^{-3} \times y^{-2})$

$= 25 \frac{1}{x^3 y^2}$

$= \frac{25}{x^3 y^2}$

Conclusion

In this article, we simplified the given expression using only positive exponents. We started by simplifying the coefficients and then simplifying the exponents. We applied the rules of exponents to simplify the expression and finally rewrote the expression with positive exponents.

Key Takeaways

• Exponents are a shorthand way of representing repeated multiplication of a number.
• The rule of subtracting the exponents when dividing like bases states that $\frac{x^m}{x^n} = x^{m-n}$.
• The rule of multiplying the exponents when raising a power to a power states that $(x^m)^n = x^{m \times n}$.
• To simplify an expression with negative exponents, we can rewrite it with positive exponents by applying the rule of multiplying the exponents.

Further Reading

If you want to learn more about exponents and simplifying expressions, I recommend checking out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponents and Exponential Functions

Practice Problems

Try simplifying the following expressions using only positive exponents:

• $\frac{9 x^2 y^3}{3 x^4 y^5}$
• $\frac{16 x^3 y^2}{4 x^5 y^4}$
• $\frac{25 x^4 y^3}{5 x^6 y^5}$
  

Understanding Exponents and Simplification

In our previous article, we discussed how to simplify expressions using only positive exponents. In this article, we will answer some frequently asked questions about simplifying expressions with positive exponents.

Q: What are exponents?

A: Exponents are a shorthand way of representing repeated multiplication of a number. For example, $x^3$ can be read as "x to the power of 3" and is equivalent to $x \times x \times x$.

Q: What is the rule for subtracting exponents when dividing like bases?

A: The rule for subtracting exponents when dividing like bases states that $\frac{x^m}{x^n} = x^{m-n}$. This means that when we divide two powers with the same base, we subtract the exponents.

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

A: The rule for multiplying exponents when raising a power to a power states that $(x^m)^n = x^{m \times n}$. This means that when we raise a power to a power, we multiply the exponents.

Q: How do I simplify an expression with negative exponents?

A: To simplify an expression with negative exponents, we can rewrite it with positive exponents by applying the rule of multiplying the exponents. For example, $x^{-3}$ can be rewritten as $\frac{1}{x^3}$.

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

A: A coefficient is a number that is multiplied by a variable, while an exponent is a power to which a variable is raised. For example, in the expression $3x^2$, 3 is the coefficient and 2 is the exponent.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, we can apply the rules of exponents to each variable separately. For example, $\frac{x^2 y^3}{x^4 y^5}$ can be simplified by applying the rule of subtracting exponents to each variable.

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

A: The order of operations when simplifying expressions with exponents is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate exponents next.
3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

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 expected value. You can also use a calculator to check your work.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions with positive exponents. We discussed the rules of exponents, how to simplify expressions with negative exponents, and how to check your work when simplifying expressions with exponents.

Key Takeaways

• Exponents are a shorthand way of representing repeated multiplication of a number.
• The rule of subtracting exponents when dividing like bases states that $\frac{x^m}{x^n} = x^{m-n}$.
• The rule of multiplying exponents when raising a power to a power states that $(x^m)^n = x^{m \times n}$.
• To simplify an expression with negative exponents, we can rewrite it with positive exponents by applying the rule of multiplying the exponents.
• The order of operations when simplifying expressions with exponents is: parentheses, exponents, multiplication and division, and addition and subtraction.

Further Reading

If you want to learn more about exponents and simplifying expressions, I recommend checking out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponents and Exponential Functions

Practice Problems

Try simplifying the following expressions using only positive exponents:

• $\frac{9 x^2 y^3}{3 x^4 y^5}$
• $\frac{16 x^3 y^2}{4 x^5 y^4}$
• $\frac{25 x^4 y^3}{5 x^6 y^5}$