Which Of The Following Is Equivalent To $10^4 \cdot 10^3$?A. $20^{12}$ B. \$10^{12}$[/tex\] C. $10^7$ D. $20^7$

When dealing with exponents, it's essential to understand the rules and properties that govern them. In this article, we will explore the concept of equivalent expressions and how to simplify them using exponent rules.

What are Exponents?

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $10^4$ means 10 multiplied by itself 4 times: $10 \times 10 \times 10 \times 10 = 10000$. Exponents can be used to represent very large or very small numbers in a more compact and manageable form.

Equivalent Expressions

Equivalent expressions are expressions that have the same value, but may be written in different forms. In the context of exponents, equivalent expressions can be obtained by applying the rules of exponent arithmetic.

The Rule of Multiplication

One of the fundamental rules of exponent arithmetic is the rule of multiplication, which states that when multiplying two numbers with the same base, you add their exponents. For example:

$$10^4 \cdot 10^3 = 10^{4+3} = 10^7$$

This rule can be applied to any numbers with the same base, not just 10.

Applying the Rule to the Given Expression

Now, let's apply the rule of multiplication to the given expression:

$$10^4 \cdot 10^3 = 10^{4+3} = 10^7$$

This means that the equivalent expression to $10^4 \cdot 10^3$ is $10^7$.

Evaluating the Answer Choices

Let's evaluate the answer choices to see which one is equivalent to $10^7$:

A. $20^{12}$

This expression has a base of 20, not 10, so it is not equivalent to $10^7$.

B. $10^{12}$

This expression has an exponent of 12, not 7, so it is not equivalent to $10^7$.

C. $10^7$

This expression has the same base and exponent as the original expression, so it is equivalent to $10^7$.

D. $20^7$

This expression has a base of 20, not 10, so it is not equivalent to $10^7$.

Conclusion

In conclusion, the equivalent expression to $10^4 \cdot 10^3$ is $10^7$. This can be obtained by applying the rule of multiplication, which states that when multiplying two numbers with the same base, you add their exponents.

Key Takeaways

• Exponents are a shorthand way of representing repeated multiplication of a number.
• Equivalent expressions are expressions that have the same value, but may be written in different forms.
• The rule of multiplication states that when multiplying two numbers with the same base, you add their exponents.
• The equivalent expression to $10^4 \cdot 10^3$ is $10^7$.

Additional Resources

For more information on exponents and equivalent expressions, check out the following resources:

• Khan Academy: Exponents and Exponential Functions
• Mathway: Exponents and Exponential Functions
• Wolfram MathWorld: Exponents and Exponential Functions
  Frequently Asked Questions (FAQs) on Exponents and Equivalent Expressions ====================================================================

In this article, we will address some of the most frequently asked questions on exponents and equivalent expressions.

Q: What is the difference between an exponent and a power?

A: An exponent is a small number that is written above and to the right of a base number, indicating how many times the base number should be multiplied by itself. A power, on the other hand, is the result of raising a base number to a certain exponent.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the following rules:

• When multiplying two numbers with the same base, add their exponents.
• When dividing two numbers with the same base, subtract their exponents.
• When raising a power to a power, multiply the exponents.

Q: What is the rule of multiplication for exponents?

A: The rule of multiplication for exponents states that when multiplying two numbers with the same base, you add their exponents. For example:

$$10^4 \cdot 10^3 = 10^{4+3} = 10^7$$

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you can follow these steps:

1. Simplify any expressions inside parentheses.
2. Evaluate any exponential expressions.
3. Multiply or divide any remaining expressions.

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

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. A negative exponent indicates that the base number should be divided by itself a certain number of times.

Q: How do I handle negative exponents?

A: To handle negative exponents, you can rewrite the expression with a positive exponent by taking the reciprocal of the base number. For example:

$$10^{-3} = \frac{1}{10^3}$$

Q: What is the rule of division for exponents?

A: The rule of division for exponents states that when dividing two numbers with the same base, you subtract their exponents. For example:

$$\frac{10^4}{10^3} = 10^{4-3} = 10^1$$

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

A: To simplify an expression with multiple exponents, you can use the following rules:

• When multiplying two numbers with the same base, add their exponents.
• When dividing two numbers with the same base, subtract their exponents.
• When raising a power to a power, multiply the exponents.

Q: What is the order of operations for exponents?

A: The order of operations for exponents is:

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any remaining expressions from left to right.
4. Addition and Subtraction: Evaluate any remaining expressions from left to right.

Conclusion

In conclusion, exponents and equivalent expressions are fundamental concepts in mathematics that can be used to simplify complex expressions. By understanding the rules of exponent arithmetic and the order of operations, you can evaluate expressions with exponents and simplify them to their simplest form.

Key Takeaways

• Exponents are a shorthand way of representing repeated multiplication of a number.
• Equivalent expressions are expressions that have the same value, but may be written in different forms.
• The rule of multiplication states that when multiplying two numbers with the same base, you add their exponents.
• The rule of division states that when dividing two numbers with the same base, you subtract their exponents.
• The order of operations for exponents is parentheses, exponents, multiplication and division, and addition and subtraction.

Additional Resources

For more information on exponents and equivalent expressions, check out the following resources:

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