Simplify The Expression: ${ \frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x} }$

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Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves rewriting complex expressions in a simpler form, often by combining like terms, canceling out common factors, or using properties of exponents. In this article, we will focus on simplifying the given expression $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$ using various techniques and properties of exponents.

Understanding the Properties of Exponents

Before we dive into simplifying the expression, it's essential to understand the properties of exponents. The properties of exponents state that:

• When multiplying two powers with the same base, we add their exponents.
• When dividing two powers with the same base, we subtract their exponents.
• When raising a power to another power, we multiply their exponents.

These properties will be crucial in simplifying the given expression.

Simplifying the Expression

To simplify the expression $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$, we can start by rewriting the expression using the properties of exponents.

$\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$

We can rewrite $25$ as $5^2$ and $125$ as $5^3$.

$\frac{5^x \cdot (5^2)^{x-1}}{5 \cdot (5^3)^x}$

Using the property of exponents that $(a^m)^n = a^{mn}$, we can simplify the expression further.

$\frac{5^x \cdot 5^{2(x-1)}}{5 \cdot 5^{3x}}$

Now, we can use the property of exponents that $a^m \cdot a^n = a^{m+n}$ to combine the exponents.

$\frac{5^x \cdot 5^{2x-2}}{5 \cdot 5^{3x}}$

Simplifying the exponents, we get:

$\frac{5^{x+2x-2}}{5 \cdot 5^{3x}}$

Combining the exponents, we get:

$\frac{5^{3x-2}}{5 \cdot 5^{3x}}$

Now, we can use the property of exponents that $\frac{a^m}{a^n} = a^{m-n}$ to simplify the expression further.

$\frac{5^{3x-2}}{5 \cdot 5^{3x}}$

Simplifying the expression, we get:

$\frac{5^{3x-2}}{5^{1+3x}}$

Using the property of exponents that $\frac{a^m}{a^n} = a^{m-n}$, we can simplify the expression further.

$5^{(3x-2)-(1+3x)}$

Simplifying the exponents, we get:

$5^{-2}$

Therefore, the simplified expression is $\boxed{5^{-2}}$.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. In this article, we simplified the expression $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$ using various techniques and properties of exponents. We rewrote the expression using the properties of exponents, combined like terms, and canceled out common factors to arrive at the simplified expression $\boxed{5^{-2}}$. This example demonstrates the importance of understanding the properties of exponents and applying them to simplify complex expressions.

Frequently Asked Questions

• What is the simplified expression of $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$?
  • The simplified expression is $\boxed{5^{-2}}$.
• How do we simplify algebraic expressions?
  • We can simplify algebraic expressions by combining like terms, canceling out common factors, and using properties of exponents.
• What are the properties of exponents?
  • The properties of exponents state that when multiplying two powers with the same base, we add their exponents, when dividing two powers with the same base, we subtract their exponents, and when raising a power to another power, we multiply their exponents.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By understanding the properties of exponents and applying them to simplify complex expressions, we can arrive at the simplified expression. This example demonstrates the importance of understanding the properties of exponents and applying them to simplify complex expressions.

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Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves rewriting complex expressions in a simpler form, often by combining like terms, canceling out common factors, or using properties of exponents. In this article, we will focus on simplifying the given expression $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$ using various techniques and properties of exponents.

Q&A: Simplifying Algebraic Expressions

Q1: What is the simplified expression of $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$?

A1: The simplified expression is $\boxed{5^{-2}}$.

Q2: How do we simplify algebraic expressions?

A2: We can simplify algebraic expressions by combining like terms, canceling out common factors, and using properties of exponents.

Q3: What are the properties of exponents?

A3: The properties of exponents state that when multiplying two powers with the same base, we add their exponents, when dividing two powers with the same base, we subtract their exponents, and when raising a power to another power, we multiply their exponents.

Q4: How do we rewrite $25$ as an exponent of $5$?

A4: We can rewrite $25$ as $5^2$.

Q5: How do we rewrite $125$ as an exponent of $5$?

A5: We can rewrite $125$ as $5^3$.

Q6: How do we simplify the expression $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$ using the properties of exponents?

A6: We can simplify the expression by rewriting $25$ as $5^2$ and $125$ as $5^3$, then using the properties of exponents to combine like terms and cancel out common factors.

Q7: What is the final simplified expression of $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$?

A7: The final simplified expression is $\boxed{5^{-2}}$.

Tips and Tricks for Simplifying Algebraic Expressions

• Always start by rewriting the expression using the properties of exponents.
• Combine like terms and cancel out common factors to simplify the expression.
• Use the properties of exponents to rewrite powers of powers and simplify the expression.
• Check your work by plugging in values for the variables to ensure that the simplified expression is correct.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By understanding the properties of exponents and applying them to simplify complex expressions, we can arrive at the simplified expression. This example demonstrates the importance of understanding the properties of exponents and applying them to simplify complex expressions.

Frequently Asked Questions

• What is the simplified expression of $\frac{5^x \cdot 25^{x-1}}{5 \cdot 125^x}$?
  • The simplified expression is $\boxed{5^{-2}}$.
• How do we simplify algebraic expressions?
  • We can simplify algebraic expressions by combining like terms, canceling out common factors, and using properties of exponents.
• What are the properties of exponents?
  • The properties of exponents state that when multiplying two powers with the same base, we add their exponents, when dividing two powers with the same base, we subtract their exponents, and when raising a power to another power, we multiply their exponents.

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. By understanding the properties of exponents and applying them to simplify complex expressions, we can arrive at the simplified expression. This example demonstrates the importance of understanding the properties of exponents and applying them to simplify complex expressions.