Simplify. Express Your Answer Using Positive Exponents.$8 G^0 \cdot 4 G \cdot 7 G$

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Understanding Exponents and Their Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this problem, we're given the expression $8 g^0 \cdot 4 g \cdot 7 g$. To simplify this expression, we need to apply the rules of exponents, specifically the rule for multiplying variables with the same base.

Rule for Multiplying Variables with the Same Base

The rule states that when multiplying variables with the same base, we add their exponents. In this case, we have $g^0$, $g$, and $g$ as the variables. Since they all have the same base, $g$, we can add their exponents.

Applying the Rule

To apply the rule, we first need to express the coefficients ($8$ and $4$) in terms of the base $g$. We can do this by rewriting them as powers of $g$. Since $8 = g^0 \cdot 8$ and $4 = g^0 \cdot 4$, we can rewrite the expression as:

$$8 g^0 \cdot 4 g \cdot 7 g = (g^0 \cdot 8) \cdot (g^0 \cdot 4) \cdot 7 g$$

Simplifying the Expression

Now that we have rewritten the coefficients in terms of the base $g$, we can simplify the expression by adding the exponents. We have:

$$(g^0 \cdot 8) \cdot (g^0 \cdot 4) \cdot 7 g = g^0 \cdot g^0 \cdot 7 g \cdot 8 \cdot 4$$

Applying the Rule for Multiplying Powers with the Same Base

Since we have the same base, $g$, we can add the exponents. We have:

$$g^0 \cdot g^0 \cdot 7 g \cdot 8 \cdot 4 = g^{0+0} \cdot 7 g \cdot 8 \cdot 4$$

Simplifying the Exponents

Now that we have added the exponents, we can simplify the expression further. We have:

$$g^{0+0} \cdot 7 g \cdot 8 \cdot 4 = g^0 \cdot 7 g \cdot 8 \cdot 4$$

Applying the Rule for Multiplying Powers with the Same Base Again

Since we have the same base, $g$, we can add the exponents. We have:

$$g^0 \cdot 7 g \cdot 8 \cdot 4 = g^0 \cdot g^{1+1} \cdot 8 \cdot 4$$

Simplifying the Exponents Again

Now that we have added the exponents, we can simplify the expression further. We have:

$$g^0 \cdot g^{1+1} \cdot 8 \cdot 4 = g^0 \cdot g^2 \cdot 8 \cdot 4$$

Applying the Rule for Multiplying Powers with the Same Base Again

Since we have the same base, $g$, we can add the exponents. We have:

$$g^0 \cdot g^2 \cdot 8 \cdot 4 = g^{0+2} \cdot 8 \cdot 4$$

Simplifying the Exponents Again

Now that we have added the exponents, we can simplify the expression further. We have:

$$g^{0+2} \cdot 8 \cdot 4 = g^2 \cdot 8 \cdot 4$$

Multiplying the Coefficients

Now that we have simplified the exponents, we can multiply the coefficients. We have:

$$g^2 \cdot 8 \cdot 4 = g^2 \cdot 32$$

Final Answer

Therefore, the simplified expression is $g^2 \cdot 32$.

Conclusion

In this problem, we applied the rules of exponents to simplify the expression $8 g^0 \cdot 4 g \cdot 7 g$. We first rewrote the coefficients in terms of the base $g$, then added the exponents, and finally multiplied the coefficients. The final answer is $g^2 \cdot 32$.

Frequently Asked Questions

• What is the rule for multiplying variables with the same base? The rule states that when multiplying variables with the same base, we add their exponents.
• How do we apply the rule for multiplying powers with the same base? We add the exponents of the powers with the same base.
• What is the final answer to the problem? The final answer is $g^2 \cdot 32$.

Key Takeaways

• When dealing with exponents, it's essential to understand the rules that govern their behavior.
• The rule for multiplying variables with the same base states that we add their exponents.
• We can apply the rule for multiplying powers with the same base by adding the exponents.
• The final answer to the problem is $g^2 \cdot 32$.
  

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Understanding Exponents and Their Rules

When dealing with exponents, it's essential to understand the rules that govern their behavior. In this problem, we're given the expression $8 g^0 \cdot 4 g \cdot 7 g$. To simplify this expression, we need to apply the rules of exponents, specifically the rule for multiplying variables with the same base.

Q&A

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

A: The rule states that when multiplying variables with the same base, we add their exponents.

Q: How do we apply the rule for multiplying powers with the same base?

A: We add the exponents of the powers with the same base.

Q: What is the final answer to the problem?

A: The final answer is $g^2 \cdot 32$.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by multiplying the coefficients.

Q: How do we multiply the coefficients?

A: We multiply the coefficients by multiplying the numbers together.

Q: What is the result of multiplying the coefficients?

A: The result of multiplying the coefficients is $32$.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by combining the variables.

Q: How do we combine the variables?

A: We combine the variables by adding their exponents.

Q: What is the result of combining the variables?

A: The result of combining the variables is $g^2$.

Q: What is the final simplified expression?

A: The final simplified expression is $g^2 \cdot 32$.

Frequently Asked Questions

• What is the rule for multiplying variables with the same base? The rule states that when multiplying variables with the same base, we add their exponents.
• How do we apply the rule for multiplying powers with the same base? We add the exponents of the powers with the same base.
• What is the final answer to the problem? The final answer is $g^2 \cdot 32$.

Key Takeaways

• When dealing with exponents, it's essential to understand the rules that govern their behavior.
• The rule for multiplying variables with the same base states that we add their exponents.
• We can apply the rule for multiplying powers with the same base by adding the exponents.
• The final answer to the problem is $g^2 \cdot 32$.

Additional Resources

• For more information on exponents and their rules, please refer to the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram Alpha: Exponents and Exponential Functions

Conclusion

In this article, we have discussed the rules of exponents and how to apply them to simplify expressions. We have also provided a Q&A section to answer common questions and provide additional resources for further learning. We hope this article has been helpful in understanding the rules of exponents and how to apply them to simplify expressions.