Simplify Each Expression. Write Your Answer Using A Negative Exponent.i) $7^{-12} \cdot 7^{-5} =$ $\square$ii) $10^{-10} \cdot 10^5 =$ $\square$iii) $\frac{3^4}{3^6} =$ $\square$v)

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Understanding Negative Exponents

Negative exponents are a fundamental concept in mathematics that can be used to simplify complex expressions. In this article, we will explore how to simplify each of the given expressions using negative exponents.

i) 7−12⋅7−5=7^{-12} \cdot 7^{-5} = □\square

To simplify the expression 7−12⋅7−57^{-12} \cdot 7^{-5}, we need to apply the rule of multiplying powers with the same base. When we multiply two powers with the same base, we add their exponents.

7−12⋅7−5=7−12−5=7−177^{-12} \cdot 7^{-5} = 7^{-12-5} = 7^{-17}

Therefore, the simplified expression is 7−177^{-17}.

ii) 10−10⋅105=10^{-10} \cdot 10^5 = □\square

To simplify the expression 10−10⋅10510^{-10} \cdot 10^5, we again apply the rule of multiplying powers with the same base. When we multiply two powers with the same base, we add their exponents.

10−10⋅105=10−10+5=10−510^{-10} \cdot 10^5 = 10^{-10+5} = 10^{-5}

Therefore, the simplified expression is 10−510^{-5}.

iii) 3436=\frac{3^4}{3^6} = □\square

To simplify the expression 3436\frac{3^4}{3^6}, we need to apply the rule of dividing powers with the same base. When we divide two powers with the same base, we subtract their exponents.

3436=34−6=3−2\frac{3^4}{3^6} = 3^{4-6} = 3^{-2}

Therefore, the simplified expression is 3−23^{-2}.

Understanding Negative Exponents in Terms of Fractions

Negative exponents can be expressed as fractions in the form 1an\frac{1}{a^n}, where aa is the base and nn is the exponent. For example, 7−177^{-17} can be expressed as 1717\frac{1}{7^{17}}.

Converting Negative Exponents to Fractions

To convert a negative exponent to a fraction, we can use the following rule:

a−n=1ana^{-n} = \frac{1}{a^n}

For example, 7−177^{-17} can be converted to a fraction as follows:

7−17=17177^{-17} = \frac{1}{7^{17}}

Simplifying Expressions with Negative Exponents

Negative exponents can be used to simplify complex expressions. By applying the rules of multiplying and dividing powers with the same base, we can simplify expressions with negative exponents.

Example 1: Simplifying 2−3⋅242^{-3} \cdot 2^4

To simplify the expression 2−3⋅242^{-3} \cdot 2^4, we apply the rule of multiplying powers with the same base.

2−3⋅24=2−3+4=212^{-3} \cdot 2^4 = 2^{-3+4} = 2^1

Therefore, the simplified expression is 212^1.

Example 2: Simplifying 5355\frac{5^3}{5^5}

To simplify the expression 5355\frac{5^3}{5^5}, we apply the rule of dividing powers with the same base.

5355=53−5=5−2\frac{5^3}{5^5} = 5^{3-5} = 5^{-2}

Therefore, the simplified expression is 5−25^{-2}.

Conclusion

In conclusion, negative exponents are a powerful tool for simplifying complex expressions. By applying the rules of multiplying and dividing powers with the same base, we can simplify expressions with negative exponents. We can also convert negative exponents to fractions using the rule a−n=1ana^{-n} = \frac{1}{a^n}. By mastering the concept of negative exponents, we can simplify a wide range of mathematical expressions.

Common Mistakes to Avoid

When working with negative exponents, it's essential to avoid common mistakes. Here are a few mistakes to watch out for:

  • Not applying the rule of multiplying powers with the same base: When multiplying powers with the same base, we add their exponents. Make sure to apply this rule correctly.
  • Not applying the rule of dividing powers with the same base: When dividing powers with the same base, we subtract their exponents. Make sure to apply this rule correctly.
  • Not converting negative exponents to fractions: Negative exponents can be expressed as fractions in the form 1an\frac{1}{a^n}. Make sure to convert negative exponents to fractions correctly.

Final Tips

Here are a few final tips to help you master the concept of negative exponents:

  • Practice, practice, practice: The more you practice working with negative exponents, the more comfortable you'll become with the concept.
  • Use online resources: There are many online resources available to help you learn about negative exponents, including video tutorials and practice problems.
  • Seek help when needed: If you're struggling with negative exponents, don't be afraid to seek help from a teacher or tutor.

By following these tips and avoiding common mistakes, you'll be well on your way to mastering the concept of negative exponents.