Calculating the Relationship Between the Two Answers

Oct 13, 2025 by miodeya.com 138 views

Introduction

In mathematics, exponents play a crucial role in representing large numbers in a compact form. Geri, a student, initially believed that the answer to one of her homework problems was $$2^3$$. However, upon reviewing her work, she discovered that the correct answer was actually $$2^9$$. This article aims to compare these two answers and provide a deeper understanding of exponents.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication. In the expression $$2^3$$, the exponent $$3$$ indicates that the base $$2$$ should be multiplied by itself $$3$$ times. This can be rewritten as $$2 \times 2 \times 2 = 8$$. Exponents make it easier to work with large numbers and simplify complex calculations.

Comparing Geri's Original and Correct Answers

Geri's original answer was $$2^3$$, which equals $$8$$. The correct answer, $$2^9$$, can be calculated by multiplying the base $$2$$ by itself $$9$$ times: $$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 512$$. This means that the correct answer is $$512$$, while Geri's original answer was $$8$$.

How Do These Two Answers Compare?

To compare these two answers, we need to understand the relationship between exponents. When we have two exponents with the same base, we can compare them by looking at their exponents. In this case, the base is $$2$$, and the exponents are $$3$$ and $$9$$. Since $$9$$ is greater than $$3$$, the correct answer $$2^9$$ is larger than Geri's original answer $$2^3$$.

Calculating the Relationship Between the Two Answers

To calculate the relationship between the two answers, we can use the concept of exponential growth. When we raise a number to a power, we are essentially multiplying it by itself that many times. In this case, we have $$2^9$$, which is equal to $$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$$. We can see that the correct answer is $$512$$, which is $$64$$ times larger than Geri's original answer $$8$$.

Conclusion

In conclusion, Geri's original answer $$2^3$$ was incorrect, and the correct answer was $$2^9$$. This means that the correct answer is $$512$$, while Geri's original answer was $$8$$. We can compare these two answers by looking at their exponents and understanding the concept of exponential growth. By calculating the relationship between the two answers, we can see that the correct answer is $$64$$ times larger than Geri's original answer.

Frequently Asked Questions

Q: What is the relationship between exponents with the same base?

A: When we have two exponents with the same base, we can compare them by looking at their exponents. The exponent with the larger value will result in a larger answer.

Q: How do we calculate the relationship between two answers with different exponents?

A: We can use the concept of exponential growth to calculate the relationship between two answers with different exponents. By multiplying the base by itself the number of times indicated by the exponent, we can determine the relationship between the two answers.

Q: What is the correct answer to Geri's homework problem?

A: The correct answer to Geri's homework problem is $$2^9$$, which equals $$512$$.

Q: How does Geri's original answer compare to the correct answer?

A: Geri's original answer $$2^3$$ is $$64$$ times smaller than the correct answer $$2^9$$.

Additional Resources

For further learning, we recommend the following resources:

Khan Academy: Exponents and Exponential Functions
Mathway: Exponents and Exponential Functions
Wolfram Alpha: Exponents and Exponential Functions

Conclusion

Introduction

In our previous article, we explored the concept of exponents and exponential functions. We compared Geri's original answer $$2^3$$ to the correct answer $$2^9$$ and discussed the relationship between exponents with the same base. In this article, we will answer some frequently asked questions about exponents and exponential functions.

Q&A

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

A: An exponent is a small number that is written above and to the right of a base number, indicating how many times the base should be multiplied by itself. A power is the result of raising a base to a certain exponent.

Q: How do I calculate the value of an exponent?

A: To calculate the value of an exponent, you can multiply the base by itself the number of times indicated by the exponent. For example, $$2^3 = 2 \times 2 \times 2 = 8$$.

Q: What is the rule for multiplying exponents with the same base?

A: When multiplying exponents with the same base, you can add the exponents together. For example, $$2^3 \times 2^4 = 2^{3+4} = 2^7$$.

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, you can subtract the exponents. For example, $$2^5 \div 2^3 = 2^{5-3} = 2^2$$.

Q: How do I calculate the value of a negative exponent?

A: A negative exponent indicates that the base should be divided by itself the number of times indicated by the exponent. For example, $$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$.

Q: What is the rule for raising an exponent to another exponent?

A: When raising an exponent to another exponent, you can multiply the exponents together. For example, $$(2³⁾4 = 2^{3 \times 4} = 2^12$$.

Q: How do I calculate the value of an exponential expression with a fractional exponent?

A: An exponential expression with a fractional exponent can be calculated by raising the base to the power of the numerator and then taking the root of the result. For example, $$2^{3/2} = \sqrt{2^3} = \sqrt{8} = 2\sqrt{2}$$.

Q: What is the rule for simplifying an exponential expression with a negative exponent?

A: An exponential expression with a negative exponent can be simplified by taking the reciprocal of the base and changing the sign of the exponent. For example, $$\frac{1}{2^{-3}} = 2^3 = 8$$.

Conclusion

In conclusion, this article has provided answers to some frequently asked questions about exponents and exponential functions. We have discussed the rules for multiplying and dividing exponents, calculating the value of negative exponents, and simplifying exponential expressions with fractional and negative exponents. By understanding these concepts, you can better solve problems and make informed decisions in mathematics and other fields.

Additional Resources

For further learning, we recommend the following resources:

Khan Academy: Exponents and Exponential Functions
Mathway: Exponents and Exponential Functions
Wolfram Alpha: Exponents and Exponential Functions

Practice Problems

To practice what you have learned, try solving the following problems:

Calculate the value of $$2^4 \times 2^3$$.
Simplify the expression $$\frac{1}{2^3}$$.
Calculate the value of $$(2³⁾4$$.
Simplify the expression $$2^{3/2}$$.
Calculate the value of $$2^{-3} \times 2^4$$.

Answer Key

$2^7 = 128$
$\frac{1}{8}$
$2^{12} = 4096$
$2\sqrt{2}$
$2^1 = 2$