Select All The Expressions That Are Equivalent To $10^5 \cdot 7^5$.A. $70^{25}$ B. $\frac{1}{70^5}$ C. $70^{10}$ D. $\frac{70^8}{70^3}$

Introduction

In mathematics, equivalent expressions are those that have the same value, even if they are written differently. In this article, we will explore the concept of equivalent expressions and apply it to the given problem of selecting expressions equivalent to $10^5 \cdot 7^5$. We will examine each option carefully and determine which ones are indeed equivalent to the given expression.

Understanding Exponents

Before we dive into the problem, let's quickly review the concept of exponents. An exponent is a small number that is raised to a power, indicating how many times the base number should be multiplied by itself. For example, $a^b$ means $a$ multiplied by itself $b$ times. In the given expression, $10^5$ means $10$ multiplied by itself $5$ times, and $7^5$ means $7$ multiplied by itself $5$ times.

Option A: $70^{25}$

Let's start by examining option A, $70^{25}$. To determine if this expression is equivalent to $10^5 \cdot 7^5$, we need to break it down. $70^{25}$ can be rewritten as $(7 \cdot 10)^{25}$. Using the property of exponents that $(ab)^c = a^c \cdot b^c$, we can rewrite this as $7^{25} \cdot 10^{25}$. However, this is not equivalent to $10^5 \cdot 7^5$, as the exponents are different. Therefore, option A is not equivalent to the given expression.

Option B: $\frac{1}{70^5}$

Next, let's examine option B, $\frac{1}{70^5}$. To determine if this expression is equivalent to $10^5 \cdot 7^5$, we need to simplify it. $\frac{1}{70^5}$ can be rewritten as $\frac{1}{(7 \cdot 10)^5}$. Using the property of exponents that $\frac{1}{a^b} = a^{-b}$, we can rewrite this as $7^{-5} \cdot 10^{-5}$. However, this is not equivalent to $10^5 \cdot 7^5$, as the exponents are different. Therefore, option B is not equivalent to the given expression.

Option C: $70^{10}$

Now, let's examine option C, $70^{10}$. To determine if this expression is equivalent to $10^5 \cdot 7^5$, we need to break it down. $70^{10}$ can be rewritten as $(7 \cdot 10)^{10}$. Using the property of exponents that $(ab)^c = a^c \cdot b^c$, we can rewrite this as $7^{10} \cdot 10^{10}$. However, this is not equivalent to $10^5 \cdot 7^5$, as the exponents are different. Therefore, option C is not equivalent to the given expression.

Option D: $\frac{70^8}{70^3}$

Finally, let's examine option D, $\frac{70^8}{70^3}$. To determine if this expression is equivalent to $10^5 \cdot 7^5$, we need to simplify it. $\frac{70^8}{70^3}$ can be rewritten as $70^{8-3}$, which is equal to $70^5$. Using the property of exponents that $(ab)^c = a^c \cdot b^c$, we can rewrite this as $7^5 \cdot 10^5$. This is indeed equivalent to the given expression, as the exponents are the same.

Conclusion

In conclusion, only option D, $\frac{70^8}{70^3}$, is equivalent to the given expression $10^5 \cdot 7^5$. This is because it can be rewritten as $7^5 \cdot 10^5$, which has the same exponents as the given expression. The other options, A, B, and C, are not equivalent to the given expression, as they have different exponents.

Key Takeaways

• Equivalent expressions have the same value, even if they are written differently.
• Exponents can be used to simplify expressions and determine their equivalence.
• The property of exponents that $(ab)^c = a^c \cdot b^c$ can be used to rewrite expressions and determine their equivalence.

Final Thoughts

Q: What is an equivalent expression?

A: An equivalent expression is a mathematical expression that has the same value as another expression, even if it is written differently.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you need to simplify them and compare their values. You can use the properties of exponents, such as the product rule and the power rule, to simplify expressions and determine their equivalence.

Q: What is the product rule for exponents?

A: The product rule for exponents states that $(ab)^c = a^c \cdot b^c$. This means that when you multiply two numbers with the same exponent, you can rewrite the expression as the product of the two numbers raised to the same exponent.

Q: What is the power rule for exponents?

A: The power rule for exponents states that $(a^b)^c = a^{bc}$. This means that when you raise a number with an exponent to another power, you can rewrite the expression as the number raised to the product of the two exponents.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you can use the product rule and the power rule to rewrite the expression in a simpler form. For example, if you have the expression $(2^3 \cdot 3^2)^4$, you can use the product rule to rewrite it as $2^{12} \cdot 3^8$, and then use the power rule to rewrite it as $2^{12} \cdot 3^{8}$.

Q: What is the difference between an equivalent expression and a similar expression?

A: An equivalent expression is a mathematical expression that has the same value as another expression, while a similar expression is a mathematical expression that has a similar form or structure, but may not have the same value.

Q: Can you give an example of an equivalent expression?

A: Yes, here is an example of an equivalent expression: $2^3 \cdot 3^2$ and $6^5$. These two expressions are equivalent because they have the same value, even though they are written differently.

Q: Can you give an example of a similar expression?

A: Yes, here is an example of a similar expression: $2^3 \cdot 3^2$ and $2^4 \cdot 3^1$. These two expressions are similar because they have a similar form or structure, but they do not have the same value.

Q: How do I apply the concept of equivalent expressions to real-world problems?

A: The concept of equivalent expressions can be applied to a wide range of real-world problems, such as finance, science, and engineering. For example, in finance, equivalent expressions can be used to calculate the value of investments or to compare the costs of different options. In science, equivalent expressions can be used to describe the behavior of physical systems or to model the growth of populations. In engineering, equivalent expressions can be used to design and optimize systems or to predict the behavior of complex systems.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include:

• Not simplifying expressions properly
• Not using the product rule and the power rule correctly
• Not checking for equivalent expressions before simplifying
• Not considering the context of the problem when simplifying expressions

Q: How can I practice working with equivalent expressions?

A: There are many ways to practice working with equivalent expressions, including:

• Solving problems and exercises in a textbook or online resource
• Working on real-world problems or projects that involve equivalent expressions
• Practicing simplifying expressions with exponents
• Exploring different types of equivalent expressions, such as algebraic and geometric equivalent expressions.