$\left(\frac{5 C Z^{-3}}{x}\right)^2 = $

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Introduction

In mathematics, exponential expressions are a fundamental concept that plays a crucial role in various fields, including algebra, calculus, and engineering. When dealing with exponential expressions, it's essential to understand the rules and properties that govern them. In this article, we will focus on simplifying exponential expressions, specifically the expression (5cz−3x)2\left(\frac{5 c z^{-3}}{x}\right)^2. We will break down the solution into manageable steps, making it easier to understand and apply.

Understanding Exponential Properties

Before we dive into the solution, let's review some essential exponential properties:

  • Power Rule: (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  • Product Rule: am⋅an=am+na^m \cdot a^n = a^{m + n}
  • Quotient Rule: aman=am−n\frac{a^m}{a^n} = a^{m - n}
  • Zero Exponent: a0=1a^0 = 1

These properties will be instrumental in simplifying the given expression.

Simplifying the Expression

Now, let's apply the properties to simplify the expression (5cz−3x)2\left(\frac{5 c z^{-3}}{x}\right)^2.

Step 1: Apply the Power Rule

Using the power rule, we can rewrite the expression as:

(5cz−3x)2=(5cz−3)2x2\left(\frac{5 c z^{-3}}{x}\right)^2 = \frac{(5 c z^{-3})^2}{x^2}

Step 2: Apply the Product Rule

Applying the product rule, we can rewrite the numerator as:

(5cz−3)2x2=52⋅c2⋅(z−3)2x2\frac{(5 c z^{-3})^2}{x^2} = \frac{5^2 \cdot c^2 \cdot (z^{-3})^2}{x^2}

Step 3: Apply the Quotient Rule

Using the quotient rule, we can rewrite the expression as:

52⋅c2⋅(z−3)2x2=52⋅c2⋅z−6x2\frac{5^2 \cdot c^2 \cdot (z^{-3})^2}{x^2} = \frac{5^2 \cdot c^2 \cdot z^{-6}}{x^2}

Step 4: Simplify the Numerator

Now, let's simplify the numerator by applying the power rule:

52⋅c2⋅z−6x2=25⋅c2⋅z−6x2\frac{5^2 \cdot c^2 \cdot z^{-6}}{x^2} = \frac{25 \cdot c^2 \cdot z^{-6}}{x^2}

Step 5: Final Simplification

Finally, let's simplify the expression by combining the terms:

25⋅c2⋅z−6x2=25c2x2z6\frac{25 \cdot c^2 \cdot z^{-6}}{x^2} = \frac{25 c^2}{x^2 z^6}

Conclusion

In this article, we simplified the exponential expression (5cz−3x)2\left(\frac{5 c z^{-3}}{x}\right)^2 using the power rule, product rule, quotient rule, and zero exponent properties. By breaking down the solution into manageable steps, we made it easier to understand and apply. Remember, simplifying exponential expressions requires a solid understanding of the underlying properties and rules. With practice and patience, you'll become proficient in simplifying even the most complex expressions.

Additional Tips and Resources

  • For more practice problems, try working with different exponential expressions, such as (ambn)p\left(\frac{a^m}{b^n}\right)^p or (cdef)g\left(\frac{c^d}{e^f}\right)^g.
  • Review the properties of exponents, including the power rule, product rule, quotient rule, and zero exponent.
  • Practice simplifying expressions with negative exponents, such as 1am\frac{1}{a^m} or 1bn\frac{1}{b^n}.
  • For a more in-depth understanding of exponential expressions, try exploring topics like logarithms, exponential functions, and calculus.

Common Mistakes to Avoid

  • Failing to apply the power rule when simplifying expressions with exponents.
  • Ignoring the product rule when simplifying expressions with multiple terms.
  • Not using the quotient rule when simplifying expressions with fractions.
  • Forgetting to simplify the numerator and denominator separately.

Q: What is the power rule for exponents?

A: The power rule for exponents states that (am)n=am⋅n(a^m)^n = a^{m \cdot n}. This means that when you raise a power to another power, you multiply the exponents.

Q: How do I apply the power rule to simplify an expression?

A: To apply the power rule, simply multiply the exponents together. For example, (23)4=23⋅4=212(2^3)^4 = 2^{3 \cdot 4} = 2^{12}.

Q: What is the product rule for exponents?

A: The product rule for exponents states that am⋅an=am+na^m \cdot a^n = a^{m + n}. This means that when you multiply two powers with the same base, you add the exponents.

Q: How do I apply the product rule to simplify an expression?

A: To apply the product rule, simply add the exponents together. For example, 23⋅24=23+4=272^3 \cdot 2^4 = 2^{3 + 4} = 2^7.

Q: What is the quotient rule for exponents?

A: The quotient rule for exponents states that aman=am−n\frac{a^m}{a^n} = a^{m - n}. This means that when you divide two powers with the same base, you subtract the exponents.

Q: How do I apply the quotient rule to simplify an expression?

A: To apply the quotient rule, simply subtract the exponents. For example, 2523=25−3=22\frac{2^5}{2^3} = 2^{5 - 3} = 2^2.

Q: What is the zero exponent rule for exponents?

A: The zero exponent rule for exponents states that a0=1a^0 = 1. This means that any number raised to the power of zero is equal to 1.

Q: How do I apply the zero exponent rule to simplify an expression?

A: To apply the zero exponent rule, simply replace the exponent with 1. For example, 20=12^0 = 1.

Q: Can I simplify an expression with a negative exponent?

A: Yes, you can simplify an expression with a negative exponent. To do this, you can rewrite the expression with a positive exponent by moving the base to the other side of the fraction.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, simply rewrite the expression with a positive exponent. For example, 123=2−3\frac{1}{2^3} = 2^{-3}.

Q: Can I simplify an expression with a fraction as an exponent?

A: Yes, you can simplify an expression with a fraction as an exponent. To do this, you can rewrite the expression with a positive exponent by multiplying the base by the reciprocal of the fraction.

Q: How do I simplify an expression with a fraction as an exponent?

A: To simplify an expression with a fraction as an exponent, simply rewrite the expression with a positive exponent. For example, (212)3=232(2^{\frac{1}{2}})^3 = 2^{\frac{3}{2}}.

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

A: Some common mistakes to avoid when simplifying exponential expressions include:

  • Failing to apply the power rule when simplifying expressions with exponents.
  • Ignoring the product rule when simplifying expressions with multiple terms.
  • Not using the quotient rule when simplifying expressions with fractions.
  • Forgetting to simplify the numerator and denominator separately.

Q: How can I practice simplifying exponential expressions?

A: You can practice simplifying exponential expressions by working with different types of expressions, such as:

  • Expressions with positive exponents, such as 23⋅242^3 \cdot 2^4
  • Expressions with negative exponents, such as 123\frac{1}{2^3}
  • Expressions with fractions as exponents, such as (212)3(2^{\frac{1}{2}})^3
  • Expressions with multiple terms, such as 23⋅342^3 \cdot 3^4

By practicing simplifying exponential expressions, you can become more confident and proficient in solving complex problems.