Rewrite The Following Without An Exponent: 9 − 2 9^{-2} 9 − 2

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Understanding Negative Exponents

In mathematics, a negative exponent is a shorthand way of expressing a fraction. It is a powerful tool that allows us to simplify complex expressions and solve equations more efficiently. In this article, we will explore how to rewrite negative exponents without using the exponent notation.

What is a Negative Exponent?

A negative exponent is a mathematical operation that involves raising a number to a power that is less than zero. For example, 9−29^{-2} is a negative exponent because the exponent -2 is less than zero. To rewrite this expression without using the exponent notation, we need to understand the concept of negative exponents and how they can be simplified.

Rewriting Negative Exponents

To rewrite a negative exponent without using the exponent notation, we can use the following formula:

a−n=1ana^{-n} = \frac{1}{a^n}

where aa is a non-zero number and nn is a positive integer.

Using this formula, we can rewrite the expression 9−29^{-2} as follows:

9−2=1929^{-2} = \frac{1}{9^2}

Simplifying the Expression

Now that we have rewritten the expression 9−29^{-2} as a fraction, we can simplify it further by evaluating the denominator.

92=9×9=819^2 = 9 \times 9 = 81

Therefore, the expression 9−29^{-2} can be simplified as follows:

9−2=1819^{-2} = \frac{1}{81}

Real-World Applications

Negative exponents have many real-world applications in mathematics and science. For example, they are used in physics to describe the behavior of particles at the atomic and subatomic level. They are also used in engineering to design and analyze complex systems.

Conclusion

In conclusion, rewriting negative exponents without using the exponent notation is a powerful tool that can simplify complex expressions and solve equations more efficiently. By understanding the concept of negative exponents and using the formula a−n=1ana^{-n} = \frac{1}{a^n}, we can rewrite expressions like 9−29^{-2} as fractions and simplify them further.

Examples of Rewriting Negative Exponents

Here are some examples of rewriting negative exponents without using the exponent notation:

  • 2−3=123=182^{-3} = \frac{1}{2^3} = \frac{1}{8}
  • 5−4=154=16255^{-4} = \frac{1}{5^4} = \frac{1}{625}
  • 3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}

Tips and Tricks

Here are some tips and tricks for rewriting negative exponents without using the exponent notation:

  • Make sure to use the formula a−n=1ana^{-n} = \frac{1}{a^n} to rewrite the expression.
  • Simplify the denominator by evaluating the exponent.
  • Use the result to solve equations or simplify complex expressions.

Common Mistakes

Here are some common mistakes to avoid when rewriting negative exponents without using the exponent notation:

  • Not using the formula a−n=1ana^{-n} = \frac{1}{a^n} to rewrite the expression.
  • Not simplifying the denominator by evaluating the exponent.
  • Not using the result to solve equations or simplify complex expressions.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about rewriting negative exponents without using the exponent 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, 9−29^{-2} is a negative exponent because the exponent -2 is less than zero.

Q: How do I rewrite a negative exponent without using the exponent notation?

A: To rewrite a negative exponent without using the exponent notation, you can use the formula a−n=1ana^{-n} = \frac{1}{a^n}, where aa is a non-zero number and nn is a positive integer.

Q: What is the formula for rewriting negative exponents?

A: The formula for rewriting negative exponents is a−n=1ana^{-n} = \frac{1}{a^n}.

Q: Can I use the formula to rewrite any negative exponent?

A: Yes, you can use the formula to rewrite any negative exponent. The formula works for any non-zero number aa and any positive integer nn.

Q: How do I simplify the expression after rewriting the negative exponent?

A: To simplify the expression after rewriting the negative exponent, you need to evaluate the denominator. For example, if you rewrite the expression 9−29^{-2} as 192\frac{1}{9^2}, you need to evaluate the denominator 929^2 to get 8181. Then, you can simplify the expression to 181\frac{1}{81}.

Q: What are some common mistakes to avoid when rewriting negative exponents?

A: Some common mistakes to avoid when rewriting negative exponents include:

  • Not using the formula a−n=1ana^{-n} = \frac{1}{a^n} to rewrite the expression.
  • Not simplifying the denominator by evaluating the exponent.
  • Not using the result to solve equations or simplify complex expressions.

Q: Can I use the formula to rewrite negative exponents with fractions?

A: Yes, you can use the formula to rewrite negative exponents with fractions. For example, if you have the expression 12−3\frac{1}{2^{-3}}, you can rewrite it as 1123=23=8\frac{1}{\frac{1}{2^3}} = 2^3 = 8.

Q: Can I use the formula to rewrite negative exponents with decimals?

A: Yes, you can use the formula to rewrite negative exponents with decimals. For example, if you have the expression 3−2.53^{-2.5}, you can rewrite it as 132.5\frac{1}{3^{2.5}}.

Q: Can I use the formula to rewrite negative exponents with variables?

A: Yes, you can use the formula to rewrite negative exponents with variables. For example, if you have the expression x−2x^{-2}, you can rewrite it as 1x2\frac{1}{x^2}.

Conclusion

In conclusion, rewriting negative exponents without using the exponent notation is a powerful tool that can simplify complex expressions and solve equations more efficiently. By understanding the concept of negative exponents and using the formula a−n=1ana^{-n} = \frac{1}{a^n}, you can rewrite expressions like 9−29^{-2} as fractions and simplify them further.