Find The Exact Values Of The Following, Giving Your Answers As Fractions.a) 3 − 2 3^{-2} 3 − 2 B) 4 − 3 4^{-3} 4 − 3 C) 2 − 6 2^{-6} 2 − 6

Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with negative exponents, it's essential to understand how to rewrite them in fractional notation. In this article, we will explore the concept of negative exponents and learn how to find the exact values of the given expressions.

What are Negative Exponents?

A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. For example, $3^{-2}$ means 3 raised to the power of -2. To understand this concept, let's consider the following:

• $a^{-n} = \frac{1}{a^n}$

This means that any number raised to a negative power can be rewritten as the reciprocal of the number raised to the positive power.

Rewriting Negative Exponents in Fractional Notation

To rewrite a negative exponent in fractional notation, we can use the following rule:

• $a^{-n} = \frac{1}{a^n}$

For example, let's rewrite $3^{-2}$ in fractional notation:

• $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

Similarly, we can rewrite $4^{-3}$ in fractional notation:

• $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$

And finally, we can rewrite $2^{-6}$ in fractional notation:

• $2^{-6} = \frac{1}{2^6} = \frac{1}{64}$

Finding the Exact Values of the Given Expressions

Now that we have rewritten the negative exponents in fractional notation, we can find the exact values of the given expressions.

a) $3^{-2}$

As we have already seen, $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

b) $4^{-3}$

Similarly, $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$.

c) $2^{-6}$

And finally, $2^{-6} = \frac{1}{2^6} = \frac{1}{64}$.

Conclusion

In this article, we have explored the concept of negative exponents and learned how to rewrite them in fractional notation. We have also found the exact values of the given expressions by applying the rules of negative exponents. By understanding these concepts, we can solve a wide range of mathematical problems involving exponents and fractions.

Key Takeaways

• Negative exponents can be rewritten in fractional notation using the rule $a^{-n} = \frac{1}{a^n}$.
• To find the exact value of a negative exponent, we can rewrite it in fractional notation and simplify the expression.
• Understanding negative exponents and fractional notation is essential for solving mathematical problems involving exponents and fractions.

Further Reading

If you want to learn more about exponents and fractions, here are some additional resources:

• Khan Academy: Exponents and Fractions
• Mathway: Exponents and Fractions
• Wolfram Alpha: Exponents and Fractions

Introduction

In our previous article, we explored the concept of negative exponents and learned how to rewrite them in fractional notation. However, we know that practice makes perfect, and the best way to reinforce our understanding of these concepts is through practice and review. In this article, we will provide a Q&A guide to help you better understand negative exponents and fractional notation.

Q: What is a negative exponent?

A: A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. For example, $3^{-2}$ means 3 raised to the power of -2.

Q: How do I rewrite a negative exponent in fractional notation?

A: To rewrite a negative exponent in fractional notation, you can use the following rule:

• $a^{-n} = \frac{1}{a^n}$

For example, let's rewrite $3^{-2}$ in fractional notation:

• $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

Q: What is the difference between a negative exponent and a positive exponent?

A: The main difference between a negative exponent and a positive exponent is the direction of the operation. A positive exponent involves repeated multiplication, while a negative exponent involves repeated division.

Q: How do I simplify a negative exponent?

A: To simplify a negative exponent, you can rewrite it in fractional notation and simplify the expression. For example, let's simplify $4^{-3}$:

• $4^{-3} = \frac{1}{4^3} = \frac{1}{64}$

Q: Can I use negative exponents with fractions?

A: Yes, you can use negative exponents with fractions. For example, let's rewrite $\frac{1}{2^{-3}}$ in fractional notation:

• $\frac{1}{2^{-3}} = 2^3 = 8$

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

A: To evaluate an expression with a negative exponent, you can rewrite it in fractional notation and simplify the expression. For example, let's evaluate $3^{-2} \cdot 4^{-3}$:

• $3^{-2} \cdot 4^{-3} = \frac{1}{3^2} \cdot \frac{1}{4^3} = \frac{1}{9} \cdot \frac{1}{64} = \frac{1}{576}$

Q: Can I use negative exponents with variables?

A: Yes, you can use negative exponents with variables. For example, let's rewrite $x^{-2}$ in fractional notation:

• $x^{-2} = \frac{1}{x^2}$

Q: How do I solve an equation with a negative exponent?

A: To solve an equation with a negative exponent, you can rewrite it in fractional notation and simplify the expression. For example, let's solve $2^{-3} = x^{-2}$:

• $2^{-3} = x^{-2} \Rightarrow \frac{1}{2^3} = \frac{1}{x^2} \Rightarrow \frac{1}{8} = \frac{1}{x^2} \Rightarrow x^2 = 8 \Rightarrow x = \pm \sqrt{8}$

Conclusion

In this article, we have provided a Q&A guide to help you better understand negative exponents and fractional notation. We have covered a range of topics, from the definition of negative exponents to solving equations with negative exponents. By practicing and reviewing these concepts, you can become proficient in working with negative exponents and fractional notation.

Key Takeaways

• Negative exponents can be rewritten in fractional notation using the rule $a^{-n} = \frac{1}{a^n}$.
• To simplify a negative exponent, you can rewrite it in fractional notation and simplify the expression.
• Negative exponents can be used with fractions, variables, and in equations.

Further Reading

If you want to learn more about negative exponents and fractional notation, here are some additional resources:

• Khan Academy: Exponents and Fractions
• Mathway: Exponents and Fractions
• Wolfram Alpha: Exponents and Fractions

By following these resources, you can deepen your understanding of negative exponents and fractional notation and become proficient in solving mathematical problems involving these concepts.