Which Statement Best Describes Jim's First Error?A. He Did Not Multiply 3 5 \frac{3}{5} 5 3 ​ By 2 Before Applying The Power.B. He Did Not Apply The Power To The Denominator Of 3 5 \frac{3}{5} 5 3 ​ .C. He Did Not Evaluate 3 3 3^3 3 3 Correctly.D.

Introduction

Exponentiation is a fundamental concept in mathematics that deals with the repeated multiplication of a number by itself. It is a crucial operation in various mathematical expressions, and understanding it is essential for solving problems in algebra, geometry, and other branches of mathematics. In this article, we will analyze a specific problem involving exponentiation and identify the mistake made by Jim in his calculation.

The Problem

Jim was asked to evaluate the expression $\left(\frac{3}{5}\right)^2 \cdot 3^3$. However, he made an error in his calculation. To determine which statement best describes Jim's first error, we need to carefully examine each option.

Option A: He did not multiply $\frac{3}{5}$ by 2 before applying the power

Let's analyze the expression $\left(\frac{3}{5}\right)^2 \cdot 3^3$. According to the order of operations, we need to evaluate the expression inside the parentheses first. In this case, we need to multiply $\frac{3}{5}$ by 2 before applying the power. However, this is not the correct interpretation of the expression. The correct interpretation is that we need to apply the power to the fraction $\frac{3}{5}$, not multiply it by 2.

Option B: He did not apply the power to the denominator of $\frac{3}{5}$

To evaluate the expression $\left(\frac{3}{5}\right)^2 \cdot 3^3$, we need to apply the power to the fraction $\frac{3}{5}$. This means that we need to raise both the numerator and the denominator to the power of 2. Therefore, the correct calculation is:

$$\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}$$

This is the correct interpretation of the expression, and it is not the mistake made by Jim.

Option C: He did not evaluate $3^3$ correctly

To evaluate the expression $\left(\frac{3}{5}\right)^2 \cdot 3^3$, we need to calculate the value of $3^3$. This means that we need to multiply 3 by itself three times:

$$3^3 = 3 \cdot 3 \cdot 3 = 27$$

This is the correct calculation, and it is not the mistake made by Jim.

Conclusion

Based on the analysis of the problem, we can conclude that Jim's first error was not related to multiplying $\frac{3}{5}$ by 2 before applying the power, not applying the power to the denominator of $\frac{3}{5}$, or not evaluating $3^3$ correctly. However, we did not identify the correct mistake made by Jim. To determine the correct answer, we need to examine the original problem and identify the specific error made by Jim.

Understanding Exponentiation

Exponentiation is a fundamental concept in mathematics that deals with the repeated multiplication of a number by itself. It is a crucial operation in various mathematical expressions, and understanding it is essential for solving problems in algebra, geometry, and other branches of mathematics.

The Rules of Exponentiation

There are several rules of exponentiation that we need to follow when evaluating expressions:

• Product of Powers Rule: When multiplying two powers with the same base, we add the exponents.
• Power of a Power Rule: When raising a power to a power, we multiply the exponents.
• Power of a Product Rule: When raising a product to a power, we raise each factor to the power.
• Zero Exponent Rule: Any non-zero number raised to the power of 0 is equal to 1.

Examples of Exponentiation

Here are some examples of exponentiation:

• $2^3 = 2 \cdot 2 \cdot 2 = 8$
• $(2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \cdot 9 = 36$
• $(2^3)^2 = 2^6 = 64$
• $2^0 = 1$

Common Mistakes in Exponentiation

There are several common mistakes that we need to avoid when evaluating expressions involving exponentiation:

• Not following the order of operations: We need to follow the order of operations when evaluating expressions involving exponentiation.
• Not applying the power to the correct base: We need to apply the power to the correct base when evaluating expressions involving exponentiation.
• Not evaluating the expression correctly: We need to evaluate the expression correctly when evaluating expressions involving exponentiation.

Conclusion

In conclusion, exponentiation is a fundamental concept in mathematics that deals with the repeated multiplication of a number by itself. It is a crucial operation in various mathematical expressions, and understanding it is essential for solving problems in algebra, geometry, and other branches of mathematics. By following the rules of exponentiation and avoiding common mistakes, we can evaluate expressions involving exponentiation correctly.

Final Answer

Based on the analysis of the problem, we can conclude that Jim's first error was not related to multiplying $\frac{3}{5}$ by 2 before applying the power, not applying the power to the denominator of $\frac{3}{5}$, or not evaluating $3^3$ correctly. However, we did not identify the correct mistake made by Jim. To determine the correct answer, we need to examine the original problem and identify the specific error made by Jim.

References

• "Algebra" by Michael Artin: This book provides a comprehensive introduction to algebra, including exponentiation.
• "Calculus" by Michael Spivak: This book provides a comprehensive introduction to calculus, including exponentiation.
• "Mathematics for Computer Science" by Eric Lehman: This book provides a comprehensive introduction to mathematics for computer science, including exponentiation.

Further Reading

For further reading on exponentiation, we recommend the following resources:

• "Exponentiation" by Wolfram MathWorld: This article provides a comprehensive introduction to exponentiation, including its rules and examples.
• "Exponentiation" by Khan Academy: This article provides a comprehensive introduction to exponentiation, including its rules and examples.
• "Exponentiation" by MIT OpenCourseWare: This article provides a comprehensive introduction to exponentiation, including its rules and examples.
  Q&A: Exponentiation =====================

Q: What is exponentiation?

A: Exponentiation is a mathematical operation that involves raising a number to a power. It is a fundamental concept in mathematics that deals with the repeated multiplication of a number by itself.

Q: What are the rules of exponentiation?

A: There are several rules of exponentiation that we need to follow when evaluating expressions:

• Product of Powers Rule: When multiplying two powers with the same base, we add the exponents.
• Power of a Power Rule: When raising a power to a power, we multiply the exponents.
• Power of a Product Rule: When raising a product to a power, we raise each factor to the power.
• Zero Exponent Rule: Any non-zero number raised to the power of 0 is equal to 1.

Q: How do I evaluate expressions involving exponentiation?

A: To evaluate expressions involving exponentiation, we need to follow the order of operations. This means that we need to evaluate the expression inside the parentheses first, then apply the power, and finally multiply the results.

Q: What are some common mistakes to avoid when evaluating expressions involving exponentiation?

A: There are several common mistakes that we need to avoid when evaluating expressions involving exponentiation:

• Not following the order of operations: We need to follow the order of operations when evaluating expressions involving exponentiation.
• Not applying the power to the correct base: We need to apply the power to the correct base when evaluating expressions involving exponentiation.
• Not evaluating the expression correctly: We need to evaluate the expression correctly when evaluating expressions involving exponentiation.

Q: How do I apply the power of a product rule?

A: To apply the power of a product rule, we need to raise each factor to the power. For example, if we have the expression $(2 \cdot 3)^2$, we need to raise each factor to the power of 2:

$$(2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \cdot 9 = 36$$

Q: How do I apply the power of a power rule?

A: To apply the power of a power rule, we need to multiply the exponents. For example, if we have the expression $(2^3)^2$, we need to multiply the exponents:

$$(2^3)^2 = 2^{3 \cdot 2} = 2^6 = 64$$

Q: What is the zero exponent rule?

A: The zero exponent rule states that any non-zero number raised to the power of 0 is equal to 1. For example, if we have the expression $2^0$, we need to evaluate it as follows:

$$2^0 = 1$$

Q: How do I evaluate expressions involving negative exponents?

A: To evaluate expressions involving negative exponents, we need to use the rule that $a^{-n} = \frac{1}{a^n}$. For example, if we have the expression $2^{-3}$, we need to evaluate it as follows:

$$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$

Q: How do I evaluate expressions involving fractional exponents?

A: To evaluate expressions involving fractional exponents, we need to use the rule that $a^{m/n} = \sqrt[n]{a^m}$. For example, if we have the expression $2^{3/4}$, we need to evaluate it as follows:

$$2^{3/4} = \sqrt[4]{2^3} = \sqrt[4]{8} = 2$$

Conclusion

In conclusion, exponentiation is a fundamental concept in mathematics that deals with the repeated multiplication of a number by itself. By following the rules of exponentiation and avoiding common mistakes, we can evaluate expressions involving exponentiation correctly.