Evaluate The Expression. Write Your Answer As A Whole Number Or As A Simplified Fraction.$\frac{3^{-9}}{3^{-5} \cdot 3^{-2}} = \frac{\square}{\square}$

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Introduction

When dealing with fractions that involve exponents, it's essential to understand the rules of exponentiation and how to simplify expressions. In this article, we will evaluate the expression 3−93−5⋅3−2\frac{3^{-9}}{3^{-5} \cdot 3^{-2}} and simplify it to a whole number or a simplified fraction.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, 343^4 means 3×3×3×33 \times 3 \times 3 \times 3. When dealing with fractions that involve exponents, we need to understand the rules of exponentiation.

Rule 1: Product of Powers

When multiplying two numbers with the same base, we add their exponents. For example, 34×32=34+2=363^4 \times 3^2 = 3^{4+2} = 3^6.

Rule 2: Quotient of Powers

When dividing two numbers with the same base, we subtract their exponents. For example, 3432=34−2=32\frac{3^4}{3^2} = 3^{4-2} = 3^2.

Evaluating the Expression

Now that we understand the rules of exponentiation, let's evaluate the expression 3−93−5⋅3−2\frac{3^{-9}}{3^{-5} \cdot 3^{-2}}.

Step 1: Simplify the Denominator

Using Rule 2, we can simplify the denominator by subtracting the exponents.

3−5⋅3−2=3−5−2=3−73^{-5} \cdot 3^{-2} = 3^{-5-2} = 3^{-7}

So, the expression becomes:

3−93−7\frac{3^{-9}}{3^{-7}}

Step 2: Simplify the Fraction

Using Rule 2, we can simplify the fraction by subtracting the exponents.

3−93−7=3−9−(−7)=3−9+7=3−2\frac{3^{-9}}{3^{-7}} = 3^{-9-(-7)} = 3^{-9+7} = 3^{-2}

Simplifying the Result

Now that we have simplified the expression, we can rewrite it as a simplified fraction.

3−2=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}

Conclusion

In this article, we evaluated the expression 3−93−5⋅3−2\frac{3^{-9}}{3^{-5} \cdot 3^{-2}} and simplified it to a whole number or a simplified fraction. We used the rules of exponentiation to simplify the expression and arrived at the final result of 19\frac{1}{9}.

Frequently Asked Questions

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

A: When multiplying two numbers with the same base, we add their exponents. For example, 34×32=34+2=363^4 \times 3^2 = 3^{4+2} = 3^6.

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

A: When dividing two numbers with the same base, we subtract their exponents. For example, 3432=34−2=32\frac{3^4}{3^2} = 3^{4-2} = 3^2.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to understand the rules of exponentiation and apply them to the expression. You can use the product of powers rule to simplify the numerator and denominator separately, and then simplify the resulting fraction.

Final Answer

The final answer is 19\boxed{\frac{1}{9}}.

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Introduction

In our previous article, we evaluated the expression 3−93−5⋅3−2\frac{3^{-9}}{3^{-5} \cdot 3^{-2}} and simplified it to a whole number or a simplified fraction. In this article, we will provide a Q&A guide to help you understand the rules of exponentiation and how to simplify expressions with exponents.

Frequently Asked Questions

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

A: When multiplying two numbers with the same base, we add their exponents. For example, 34×32=34+2=363^4 \times 3^2 = 3^{4+2} = 3^6.

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

A: When dividing two numbers with the same base, we subtract their exponents. For example, 3432=34−2=32\frac{3^4}{3^2} = 3^{4-2} = 3^2.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to understand the rules of exponentiation and apply them to the expression. You can use the product of powers rule to simplify the numerator and denominator separately, and then simplify the resulting fraction.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, we multiply the exponents. For example, (34)2=34×2=38(3^4)^2 = 3^{4 \times 2} = 3^8.

Q: What is the rule for raising a power to a power with a negative exponent?

A: When raising a power to a power with a negative exponent, we multiply the exponents and change the sign of the result. For example, (3−4)2=3−4×2=3−8(3^{-4})^2 = 3^{-4 \times 2} = 3^{-8}.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you need to understand the rules of exponentiation and apply them to the expression. You can use the product of powers rule to simplify the numerator and denominator separately, and then simplify the resulting fraction.

Q: What is the rule for simplifying a fraction with exponents?

A: When simplifying a fraction with exponents, you need to understand the rules of exponentiation and apply them to the expression. You can use the product of powers rule to simplify the numerator and denominator separately, and then simplify the resulting fraction.

Advanced Topics

Q: What is the rule for simplifying a fraction with a negative exponent in the numerator?

A: When simplifying a fraction with a negative exponent in the numerator, you need to change the sign of the exponent and simplify the resulting fraction. For example, 3−432=3−4−2=3−6\frac{3^{-4}}{3^2} = 3^{-4-2} = 3^{-6}.

Q: What is the rule for simplifying a fraction with a negative exponent in the denominator?

A: When simplifying a fraction with a negative exponent in the denominator, you need to change the sign of the exponent and simplify the resulting fraction. For example, 343−2=34+2=36\frac{3^4}{3^{-2}} = 3^{4+2} = 3^6.

Conclusion

In this article, we provided a Q&A guide to help you understand the rules of exponentiation and how to simplify expressions with exponents. We covered topics such as multiplying and dividing numbers with the same base, raising a power to a power, and simplifying fractions with exponents.

Final Answer

The final answer is 19\boxed{\frac{1}{9}}.