Simplify The Expression Using Only Positive Exponents.$ W^3 \cdot W^{-9} $The Simplified Expression Is $ \square $.

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Introduction

When dealing with exponents, it's essential to understand the rules of exponentiation to simplify expressions. In this article, we will focus on simplifying the expression w3⋅w−9w^3 \cdot w^{-9} using only positive exponents.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication. For example, w3w^3 can be written as w⋅w⋅ww \cdot w \cdot w. When we multiply two numbers with the same base, we add their exponents. However, when we multiply two numbers with different bases, we cannot simply add their exponents.

The Rule for Multiplying Exponents with the Same Base

When we multiply two numbers with the same base, we add their exponents. This rule is represented as:

am⋅an=am+na^m \cdot a^n = a^{m+n}

where aa is the base and mm and nn are the exponents.

The Rule for Multiplying Exponents with Different Bases

When we multiply two numbers with different bases, we cannot simply add their exponents. Instead, we multiply the bases and add the exponents.

Simplifying the Expression

Now that we understand the rules for multiplying exponents, let's simplify the expression w3⋅w−9w^3 \cdot w^{-9}.

To simplify this expression, we need to apply the rule for multiplying exponents with the same base. Since both terms have the same base ww, we can add their exponents.

w3⋅w−9=w3+(−9)w^3 \cdot w^{-9} = w^{3+(-9)}

Applying the Rule for Negative Exponents

When we add a negative exponent to a positive exponent, we can rewrite the expression with a positive exponent. To do this, we need to apply the rule for negative exponents.

a−n=1ana^{-n} = \frac{1}{a^n}

Using this rule, we can rewrite the expression as:

w3+(−9)=w3−9w^{3+(-9)} = w^{3-9}

Simplifying the Expression Further

Now that we have rewritten the expression with a positive exponent, we can simplify it further.

w3−9=w−6w^{3-9} = w^{-6}

Applying the Rule for Negative Exponents Again

Since we have a negative exponent, we can apply the rule for negative exponents again to rewrite the expression with a positive exponent.

w−6=1w6w^{-6} = \frac{1}{w^6}

The Final Answer

The simplified expression is 1w6\boxed{\frac{1}{w^6}}.

Conclusion

In this article, we simplified the expression w3⋅w−9w^3 \cdot w^{-9} using only positive exponents. We applied the rules for multiplying exponents with the same base and negative exponents to arrive at the final answer. This article demonstrates the importance of understanding the rules of exponentiation to simplify expressions.

Frequently Asked Questions

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

A: The rule for multiplying exponents with the same base is am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the rule for multiplying exponents with different bases?

A: The rule for multiplying exponents with different bases is am⋅bn=am⋅bna^m \cdot b^n = a^m \cdot b^n.

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

A: To simplify an expression with a negative exponent, you can apply the rule for negative exponents: a−n=1ana^{-n} = \frac{1}{a^n}.

Q: What is the final answer to the expression w3⋅w−9w^3 \cdot w^{-9}?

A: The final answer to the expression w3⋅w−9w^3 \cdot w^{-9} is 1w6\frac{1}{w^6}.

References

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

In our previous article, we simplified the expression w3⋅w−9w^3 \cdot w^{-9} using only positive exponents. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

Q&A

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

A: The rule for multiplying exponents with the same base is am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the rule for multiplying exponents with different bases?

A: The rule for multiplying exponents with different bases is am⋅bn=am⋅bna^m \cdot b^n = a^m \cdot b^n.

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

A: To simplify an expression with a negative exponent, you can apply the rule for negative exponents: a−n=1ana^{-n} = \frac{1}{a^n}.

Q: What is the final answer to the expression w3⋅w−9w^3 \cdot w^{-9}?

A: The final answer to the expression w3⋅w−9w^3 \cdot w^{-9} is 1w6\frac{1}{w^6}.

Q: Can I simplify an expression with a zero exponent?

A: Yes, you can simplify an expression with a zero exponent. The rule for a zero exponent is a0=1a^0 = 1.

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

A: The rule for dividing exponents with the same base is am÷an=am−na^m \div a^n = a^{m-n}.

Q: What is the rule for dividing exponents with different bases?

A: The rule for dividing exponents with different bases is am÷bn=ambna^m \div b^n = \frac{a^m}{b^n}.

Q: How do I simplify an expression with a fractional exponent?

A: To simplify an expression with a fractional exponent, you can apply the rule for fractional exponents: am/n=amna^{m/n} = \sqrt[n]{a^m}.

Q: What is the final answer to the expression x1/2⋅x1/3x^{1/2} \cdot x^{1/3}?

A: The final answer to the expression x1/2⋅x1/3x^{1/2} \cdot x^{1/3} is x1/6x^{1/6}.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with exponents. We covered topics such as multiplying and dividing exponents, simplifying expressions with negative and zero exponents, and simplifying expressions with fractional exponents.

Frequently Asked Questions (FAQs)

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

A: The rule for multiplying exponents with the same base is am⋅an=am+na^m \cdot a^n = a^{m+n}.

Q: What is the rule for multiplying exponents with different bases?

A: The rule for multiplying exponents with different bases is am⋅bn=am⋅bna^m \cdot b^n = a^m \cdot b^n.

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

A: To simplify an expression with a negative exponent, you can apply the rule for negative exponents: a−n=1ana^{-n} = \frac{1}{a^n}.

Q: What is the final answer to the expression w3⋅w−9w^3 \cdot w^{-9}?

A: The final answer to the expression w3⋅w−9w^3 \cdot w^{-9} is 1w6\frac{1}{w^6}.

References

Note: The references provided are for educational purposes only and are not intended to be a comprehensive list of resources.