Simplify. Express Your Answer Using Positive Exponents.${ 3d \cdot 4d \cdot 10d }${ \square\$}

=====================================================

Understanding Exponents and Simplification

In mathematics, exponents are a shorthand way of representing repeated multiplication. For instance, the expression $3^4$ can be read as "3 to the power of 4" and is equivalent to $3 \cdot 3 \cdot 3 \cdot 3$. When simplifying expressions involving exponents, it's essential to understand the rules of exponentiation and how to apply them to simplify complex expressions.

The Rule of Multiplication with Exponents

When multiplying two or more numbers with the same base, we can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$. This rule can be applied to simplify expressions involving multiple variables with the same base.

Simplifying the Given Expression

The given expression is $3d \cdot 4d \cdot 10d$. To simplify this expression using positive exponents, we need to apply the rule of multiplication with exponents.

Breaking Down the Expression

First, let's break down the expression into its individual components:

• $3d$ can be written as $3^1 \cdot d^1$
• $4d$ can be written as $4^1 \cdot d^1$
• $10d$ can be written as $10^1 \cdot d^1$

Applying the Rule of Multiplication with Exponents

Now, let's apply the rule of multiplication with exponents to simplify the expression:

$3^1 \cdot d^1 \cdot 4^1 \cdot d^1 \cdot 10^1 \cdot d^1$

Using the rule of multiplication with exponents, we can add the exponents of the variables with the same base:

$3^1 \cdot 4^1 \cdot 10^1 \cdot d^{1+1+1}$

Simplifying the Expression

Now, let's simplify the expression by evaluating the exponents:

$3^1 \cdot 4^1 \cdot 10^1 \cdot d^3$

Using the rule of multiplication with exponents, we can multiply the coefficients:

$3 \cdot 4 \cdot 10 \cdot d^3$

Evaluating the Coefficients

Now, let's evaluate the coefficients:

$3 \cdot 4 = 12$ $12 \cdot 10 = 120$

Simplifying the Expression

Now, let's simplify the expression:

$120 \cdot d^3$

Writing the Expression in Terms of Positive Exponents

Finally, let's write the expression in terms of positive exponents:

$120d^3$

The final answer is $\boxed{120d^3}$.

Conclusion

In this article, we learned how to simplify expressions using positive exponents. We applied the rule of multiplication with exponents to simplify the given expression and evaluated the coefficients to obtain the final answer. By understanding the rules of exponentiation and how to apply them to simplify complex expressions, we can simplify expressions involving multiple variables with the same base.

Frequently Asked Questions

• What is the rule of multiplication with exponents? The rule of multiplication with exponents states that when multiplying two or more numbers with the same base, we can add their exponents.
• How do I simplify expressions involving multiple variables with the same base? To simplify expressions involving multiple variables with the same base, apply the rule of multiplication with exponents and add the exponents of the variables with the same base.
• What is the final answer to the given expression? The final answer to the given expression is $120d^3$.

Further Reading

• Exponents and Powers: A Comprehensive Guide
• Simplifying Expressions with Exponents: A Step-by-Step Guide
• Multiplication with Exponents: A Rule-Based Approach
  

=====================================================

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to simplifying expressions using positive exponents.

Q: What is the rule of multiplication with exponents?

A: The rule of multiplication with exponents states that when multiplying two or more numbers with the same base, we can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: How do I simplify expressions involving multiple variables with the same base?

A: To simplify expressions involving multiple variables with the same base, apply the rule of multiplication with exponents and add the exponents of the variables with the same base. For example, $3d \cdot 4d \cdot 10d$ can be simplified as $3^1 \cdot 4^1 \cdot 10^1 \cdot d^{1+1+1} = 120d^3$.

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

A: A positive exponent represents a power of a number, while a negative exponent represents a reciprocal of a power of a number. For example, $a^m$ represents a power of $a$, while $a^{-m}$ represents a reciprocal of a power of $a$.

Q: How do I simplify expressions involving negative exponents?

A: To simplify expressions involving negative exponents, we can use the rule of negative exponents, which states that $a^{-m} = \frac{1}{a^m}$. For example, $\frac{1}{a^m} = a^{-m}$.

Q: Can I simplify expressions involving fractions with exponents?

A: Yes, you can simplify expressions involving fractions with exponents by applying the rule of multiplication with exponents and the rule of negative exponents. For example, $\frac{a^m}{a^n} = a^{m-n}$.

Q: How do I simplify expressions involving exponents with different bases?

A: To simplify expressions involving exponents with different bases, we need to apply the rule of multiplication with exponents and the rule of negative exponents separately for each base. For example, $a^m \cdot b^n = a^m \cdot b^n$.

Q: What is the final answer to the given expression?

A: The final answer to the given expression $3d \cdot 4d \cdot 10d$ is $120d^3$.

Additional Tips and Tricks

• When simplifying expressions involving exponents, always apply the rule of multiplication with exponents and the rule of negative exponents separately for each base.
• When simplifying expressions involving fractions with exponents, apply the rule of multiplication with exponents and the rule of negative exponents separately for each base.
• When simplifying expressions involving exponents with different bases, apply the rule of multiplication with exponents and the rule of negative exponents separately for each base.

Conclusion

In this article, we addressed some of the most frequently asked questions related to simplifying expressions using positive exponents. We also provided additional tips and tricks for simplifying expressions involving exponents. By understanding the rules of exponentiation and how to apply them to simplify complex expressions, we can simplify expressions involving multiple variables with the same base.

Further Reading

• Exponents and Powers: A Comprehensive Guide
• Simplifying Expressions with Exponents: A Step-by-Step Guide
• Multiplication with Exponents: A Rule-Based Approach

Related Articles

• Simplify Expressions Using Positive Exponents
• Simplifying Expressions with Exponents: A Step-by-Step Guide
• Multiplication with Exponents: A Rule-Based Approach

Comments

We hope this article has been helpful in addressing some of the most frequently asked questions related to simplifying expressions using positive exponents. If you have any further questions or comments, please feel free to leave them below.