Consider The Following Expression: $\[ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} \\]Select Equivalent Or Not Equivalent To Indicate Whether The Expression Above Is Equivalent To The Values Or Expressions In The Last

Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When dealing with exponents, it's essential to understand the properties and rules that govern their behavior. In this article, we will explore the concept of equivalent expressions and how to determine whether two expressions are equivalent or not.

What are Equivalent Expressions?

Equivalent expressions are mathematical expressions that have the same value or result, even if they are written differently. In other words, equivalent expressions are expressions that can be simplified to the same value or result. For example, the expressions 2 × 3 and 6 are equivalent because they both equal 6.

The Given Expression

The given expression is ${ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} }$. To determine whether this expression is equivalent to the values or expressions in the last discussion category, we need to simplify it.

Simplifying the Expression

To simplify the expression, we can start by rewriting 49 as $7^2$. This gives us:

$${ 7^{\frac{1}{5}} \cdot (7^2)^{\frac{7}{5}} }$$

Using the property of exponents that states $(a^m)^n = a^{mn}$, we can rewrite the expression as:

$${ 7^{\frac{1}{5}} \cdot 7^{\frac{14}{5}} }$$

Now, we can use the property of exponents that states $a^m \cdot a^n = a^{m+n}$ to combine the two terms:

$${ 7^{\frac{1}{5} + \frac{14}{5}} }$$

Simplifying the exponent, we get:

$${ 7^{\frac{15}{5}} }$$

$${ 7^3 }$$

Is the Expression Equivalent?

Now that we have simplified the expression, we can determine whether it is equivalent to the values or expressions in the last discussion category. The simplified expression is $7^3$, which equals 343.

Conclusion

In conclusion, the expression ${ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} }$ is equivalent to the value 343. This is because the expression can be simplified to $7^3$, which equals 343.

Final Answer

The final answer is: Equivalent

Why is this expression equivalent?

This expression is equivalent because it can be simplified to a single value, 343. The properties of exponents allow us to rewrite the expression in a simpler form, which ultimately leads to the same value.

What are the properties of exponents?

The properties of exponents are a set of rules that govern how exponents behave. Some of the key properties of exponents include:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Zero Exponent: $a^0 = 1$
• Negative Exponent: $a^{-m} = \frac{1}{a^m}$

How do these properties help us simplify expressions?

These properties help us simplify expressions by allowing us to rewrite them in a simpler form. For example, the product of powers property allows us to combine two terms with the same base by adding their exponents.

What are some common mistakes to avoid when working with exponents?

Some common mistakes to avoid when working with exponents include:

• Forgetting to distribute the exponent: When multiplying two terms with exponents, it's essential to distribute the exponent to each term.
• Not using the correct order of operations: When simplifying expressions, it's essential to follow the order of operations (PEMDAS) to ensure that the expression is simplified correctly.
• Not checking for equivalent expressions: When simplifying expressions, it's essential to check if the expression is equivalent to a simpler form.

Conclusion

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Introduction

In our previous article, we explored the concept of equivalent expressions and how to determine whether two expressions are equivalent or not. We also simplified the expression ${ 7^{\frac{1}{5}} \cdot 49^{\frac{7}{5}} }$ to $7^3$, which equals 343. In this article, we will answer some frequently asked questions about 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 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 properties of exponents, such as the product of powers, power of a power, zero exponent, and negative exponent. You can also use the order of operations (PEMDAS) to ensure that the expression is simplified correctly.

Q: What is the product of powers property?

A: The product of powers property states that when you multiply two terms with the same base, you can add their exponents. For example, $a^m \cdot a^n = a^{m+n}$.

Q: What is the power of a power property?

A: The power of a power property states that when you raise a power to another power, you can multiply the exponents. For example, $(a^m)^n = a^{mn}$.

Q: What is the zero exponent property?

A: The zero exponent property states that any number raised to the power of zero is equal to 1. For example, $a^0 = 1$.

Q: What is the negative exponent property?

A: The negative exponent property states that any number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$.

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

A: To determine whether two expressions are equivalent, you can simplify each expression and compare the results. If the two expressions simplify to the same value, then they are equivalent.

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

A: Some common mistakes to avoid when working with exponents include:

• Forgetting to distribute the exponent
• Not using the correct order of operations
• Not checking for equivalent expressions

Q: How do I use exponents to solve real-world problems?

A: Exponents can be used to solve a wide range of real-world problems, such as calculating interest rates, determining the growth of populations, and modeling the spread of diseases.

Q: What are some examples of real-world problems that involve exponents?

A: Some examples of real-world problems that involve exponents include:

• Calculating the growth of a population over time
• Determining the interest rate on a loan
• Modeling the spread of a disease
• Calculating the area of a circle

Conclusion

In conclusion, exponents are a fundamental concept in mathematics that can be used to solve a wide range of real-world problems. By understanding the properties of exponents and how to simplify expressions, you can determine whether two expressions are equivalent and solve complex problems.