Simplify. Express Your Answer Using Positive Exponents.$\[5 V^3 W^3 \cdot 8 V^6 W \cdot 5 V W\\]\[$\square\$\]

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Understanding Exponents and Their Rules

Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number. In this article, we will explore the rules of exponents and how to simplify expressions using positive exponents.

What are Exponents?

Exponents are a shorthand way of writing repeated multiplication. For example, the expression 232^3 can be read as "2 to the power of 3" and is equivalent to 2×2×22 \times 2 \times 2. Exponents are used to represent the number of times a base number is multiplied by itself.

Rules of Exponents

There are several rules of exponents that we need to understand in order to simplify expressions:

  • Product of Powers Rule: When multiplying two numbers with the same base, we add their exponents. For example, am⋅an=am+na^m \cdot a^n = a^{m+n}.
  • Power of a Power Rule: When raising a power to another power, we multiply the exponents. For example, (am)n=am⋅n(a^m)^n = a^{m \cdot n}.
  • Quotient of Powers Rule: When dividing two numbers with the same base, we subtract their exponents. For example, aman=am−n\frac{a^m}{a^n} = a^{m-n}.

Simplifying the Given Expression

Now that we have a good understanding of exponents and their rules, let's simplify the given expression:

5v3w3⋅8v6w⋅5vw{5 v^3 w^3 \cdot 8 v^6 w \cdot 5 v w}

Step 1: Break Down the Expression

The first step in simplifying the expression is to break it down into its individual components. We can rewrite the expression as:

5v3w3⋅8v6w⋅5vw=(5⋅8⋅5)⋅(v3⋅v6⋅v)⋅(w3⋅w){5 v^3 w^3 \cdot 8 v^6 w \cdot 5 v w = (5 \cdot 8 \cdot 5) \cdot (v^3 \cdot v^6 \cdot v) \cdot (w^3 \cdot w)}

Step 2: Simplify the Coefficients

The next step is to simplify the coefficients. We can multiply the numbers together:

5⋅8⋅5=200{5 \cdot 8 \cdot 5 = 200}

So, the expression becomes:

200⋅(v3⋅v6⋅v)⋅(w3⋅w){200 \cdot (v^3 \cdot v^6 \cdot v) \cdot (w^3 \cdot w)}

Step 3: Simplify the Variables

Now, let's simplify the variables. We can use the product of powers rule to add the exponents:

v3⋅v6⋅v=v3+6+1=v10{v^3 \cdot v^6 \cdot v = v^{3+6+1} = v^{10}}

And we can also simplify the second set of variables:

w3⋅w=w3+1=w4{w^3 \cdot w = w^{3+1} = w^4}

So, the expression becomes:

200⋅v10⋅w4{200 \cdot v^{10} \cdot w^4}

Step 4: Write the Final Answer

The final step is to write the answer in the required format. We can rewrite the expression as:

200v10w4{200 v^{10} w^4}

This is the simplified expression using positive exponents.

Conclusion

In this article, we learned how to simplify expressions using positive exponents. We explored the rules of exponents and applied them to a given expression. By breaking down the expression into its individual components, simplifying the coefficients and variables, and writing the final answer in the required format, we were able to simplify the expression and write it in a more concise and readable form.

Key Takeaways

  • Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number.
  • There are several rules of exponents that we need to understand in order to simplify expressions, including the product of powers rule, power of a power rule, and quotient of powers rule.
  • By applying these rules and simplifying the coefficients and variables, we can simplify expressions using positive exponents.

Final Answer

The final answer is: 200v10w4\boxed{200 v^{10} w^4}

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about simplifying expressions using positive exponents.

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

A: A positive exponent represents repeated multiplication of a number, while a negative exponent represents repeated division of a number. For example, a3a^3 represents a×a×aa \times a \times a, while a−3a^{-3} represents 1a×a×a\frac{1}{a \times a \times a}.

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

A: To simplify an expression with a negative exponent, we can use the rule that a−n=1ana^{-n} = \frac{1}{a^n}. For example, a−3=1a3a^{-3} = \frac{1}{a^3}.

Q: Can I simplify an expression with a variable in the exponent?

A: Yes, you can simplify an expression with a variable in the exponent. For example, a2xa^{2x} can be simplified to (a2)x(a^2)^x.

Q: How do I simplify an expression with multiple variables in the exponent?

A: To simplify an expression with multiple variables in the exponent, we can use the product of powers rule. For example, a2⋅b3=(a2⋅b3)a^2 \cdot b^3 = (a^2 \cdot b^3).

Q: Can I simplify an expression with a coefficient in the exponent?

A: Yes, you can simplify an expression with a coefficient in the exponent. For example, 2a32a^3 can be simplified to 2(a3)2(a^3).

Q: How do I simplify an expression with a fraction in the exponent?

A: To simplify an expression with a fraction in the exponent, we can use the rule that (am)n=am⋅n(a^m)^n = a^{m \cdot n}. For example, (a12)3=a32(a^{\frac{1}{2}})^3 = a^{\frac{3}{2}}.

Common Mistakes to Avoid

When simplifying expressions using positive exponents, there are several common mistakes to avoid:

  • Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when simplifying expressions.
  • Not using the correct rules of exponents: Make sure to use the correct rules of exponents, such as the product of powers rule and the power of a power rule.
  • Not simplifying the coefficients: Make sure to simplify the coefficients in the expression.
  • Not simplifying the variables: Make sure to simplify the variables in the expression.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions using positive exponents:

  • Use the product of powers rule: The product of powers rule is a powerful tool for simplifying expressions. Make sure to use it whenever possible.
  • Use the power of a power rule: The power of a power rule is another useful tool for simplifying expressions. Make sure to use it whenever possible.
  • Simplify the coefficients: Make sure to simplify the coefficients in the expression.
  • Simplify the variables: Make sure to simplify the variables in the expression.

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying expressions using positive exponents. We also discussed common mistakes to avoid and provided tips and tricks to help you simplify expressions using positive exponents.

Key Takeaways

  • Exponents are a fundamental concept in mathematics, used to represent repeated multiplication of a number.
  • There are several rules of exponents that we need to understand in order to simplify expressions, including the product of powers rule, power of a power rule, and quotient of powers rule.
  • By following the order of operations, using the correct rules of exponents, simplifying the coefficients and variables, and using the product of powers rule and power of a power rule, we can simplify expressions using positive exponents.

Final Answer

The final answer is: 200v10w4\boxed{200 v^{10} w^4}