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Chapter 6: Problem 9

Classify each equation as true or false. If false, explain why and change the right side of the equation to make it true. a. \(\left(3 x^{2}\right)^{3}=9 x^{6}\) b. \(3^{2} \cdot 2^{3}=6^{5}\) c. \(2 x^{-2}=\frac{1}{2 x^{2}}\) d. \(\left(\frac{x^{2}}{y^{3}}\right)^{3}=\frac{x^{3}}{y^{6}}\)

Short Answer

Expert verified
All equations are false: a) should be \( 27x^{6} \); b) should be 72; c) should be \( \frac{2}{x^{2}} \); d) should be \( \frac{x^{6}}{y^{9}} \).

Step by step solution

01

Analyze Equation a

The equation is \( \left(3x^{2}\right)^{3} = 9x^{6} \). Apply the power of a power rule which states \( (a^m)^n = a^{m \cdot n} \). Hence, \( \left(3x^{2}\right)^{3} = (3)^{3} \cdot (x^{2})^{3} = 27x^{6} \), which shows that the given equation is false. Make it true by changing the right side to \( 27x^{6} \).
02

Analyze Equation b

The given equation is \( 3^{2} \cdot 2^{3} = 6^{5} \). Calculate each part separately: \( 3^{2} = 9 \) and \( 2^{3} = 8 \), which gives \( 9 \cdot 8 = 72 \). On the other hand, \( 6^{5} = 7776 \). Since 72 is not equal to 7776, the equation is false. The correct right-hand side should be \( 72 \).
03

Analyze Equation c

The equation \( 2x^{-2} = \frac{1}{2x^{2}} \) involves dealing with negative exponents. Recall that \( a^{-m} = \frac{1}{a^m} \), so \( x^{-2} = \frac{1}{x^{2}} \). Hence, \( 2x^{-2} = \frac{2}{x^{2}} \) is not equal to \( \frac{1}{2x^{2}} \). Therefore, this equation is false and should be \( 2x^{-2} = \frac{2}{x^{2}} \).
04

Analyze Equation d

The equation \( \left(\frac{x^{2}}{y^{3}}\right)^{3} = \frac{x^{3}}{y^{6}} \) can be rewritten by applying the power to each term inside the parenthesis separately: \( \left(\frac{x^{2}}{y^{3}}\right)^{3} = \frac{(x^{2})^{3}}{(y^{3})^{3}} = \frac{x^{6}}{y^{9}} \). Therefore, the equation is false and should be \( \frac{x^{6}}{y^{9}} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power of a Power Rule
The **power of a power rule** in algebra is essential for simplifying expressions with exponents. This rule states that when you raise a power to another power, you multiply the exponents together. In mathematical terms, it can be expressed as
  • \( (a^m)^n = a^{m \cdot n} \)
For example, consider the expression \( (3x^2)^3 \). When applying the power of a power rule:
  • The numerical coefficient \( 3 \) is raised to the power, resulting in \( 3^3 = 27 \).
  • The variable part \( (x^2)^3 \) simplifies to \( x^{2 \cdot 3} = x^6 \).
Therefore, the entire expression becomes \( 27x^6 \), demonstrating that correct application is crucial to ensure equation accuracy.
Negative Exponents
**Negative exponents** can seem tricky at first, but they simplify expressions by transforming multiplication into division. If you encounter a negative exponent, the rule to remember is:
  • \( a^{-m} = \frac{1}{a^m} \)
This change transforms the entire expression into a reciprocal form. For instance, if we have \( 2x^{-2} \):
  • The negative exponent means \( x^{-2} \) turns into \( \frac{1}{x^2} \).
  • Thus, \( 2x^{-2} \) becomes \( \frac{2}{x^2} \).
  • This differs from \( \frac{1}{2x^2} \), which is why it's important to carefully handle negative exponents to avoid mistakes in simplification.
Equation Analysis
**Equation analysis** involves thoroughly evaluating each side of an equation to determine its truthfulness. When analyzing, ensure each step logically follows from the previous one.
  • Consider mathematical operations including addition, subtraction, multiplication, division, and exponentiation.
  • Check for equivalent transformations to simplify both sides to a more comparable form.
  • If after simplifying, both sides are equal, the equation holds true; otherwise, it is false.
In the context of the examples, methods like calculating individual exponents and comparing results number by number are practical approaches. For instance, the equation \( 3^2 \cdot 2^3 = 6^5 \) was false but after analyzing components, we found the correct result should be \( 72 \), not \( 7776 \).
Mathematical Correction
**Mathematical correction** is necessary when an equation initially appears incorrect. Through detailed analysis and understanding of mathematical principles, we can propose changes to rectify errors. Here's how to do it effectively:
  • Use algebraic principles like the power of a power rule or negative exponents to identify missteps.
  • Adjust components of the equation that seem miscalculated or incorrectly simplified.
  • Verify each correction by recalculating and confirming consistency on both sides of the equation.
As an example, take the equation \( \left(\frac{x^2}{y^3}\right)^3 = \frac{x^3}{y^6} \). With correction, it should be \( \frac{x^6}{y^9} \), showcasing how revision can lead to true and rich mathematical statements.

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