What Is The Correct Expanded Form And Value Of $\left(\frac{1}{5}\right)^4$?A. $\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5}$B. $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} =

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 correct expanded form and value of expressions involving exponents. In this article, we will explore the correct expanded form and value of $\left(\frac{1}{5}\right)^4$.

What is an Exponent?

An exponent is a small number that is placed above and to the right of a number or expression. It represents the number of times the base is multiplied by itself. For example, in the expression $2^3$, the exponent 3 indicates that the base 2 is multiplied by itself 3 times: $2^3 = 2 \cdot 2 \cdot 2 = 8$.

Understanding the Concept of Expanded Form

The expanded form of an expression is the expression written out in full, with each term separated by a plus sign. For example, the expanded form of $2^3$ is $2 \cdot 2 \cdot 2$. In the case of $\left(\frac{1}{5}\right)^4$, the expanded form would involve multiplying the fraction $\frac{1}{5}$ by itself 4 times.

The Correct Expanded Form of $\left(\frac{1}{5}\right)^4$

The correct expanded form of $\left(\frac{1}{5}\right)^4$ is $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$. This can be written as:

$$\left(\frac{1}{5}\right)^4 = \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$$

Evaluating the Expression

To evaluate the expression, we can multiply the fractions together:

$$\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{25}$$

Comparing with the Given Options

Now that we have evaluated the expression, let's compare it with the given options:

A. $\frac{1}{5} + \frac{1}{5} + \frac{1}{5} + \frac{1}{5} = \frac{4}{5}$

B. $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{625}$

It's clear that option A is incorrect, as the sum of the fractions is not equal to $\frac{4}{5}$. Option B is also incorrect, as the product of the fractions is not equal to $\frac{1}{625}$.

Conclusion

In conclusion, the correct expanded form and value of $\left(\frac{1}{5}\right)^4$ is $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{625}$. This is the only option that accurately represents the value of the expression.

Frequently Asked Questions

• What is the correct expanded form of $\left(\frac{1}{5}\right)^4$?
• How do I evaluate the expression $\left(\frac{1}{5}\right)^4$?
• What is the value of $\left(\frac{1}{5}\right)^4$?

Answer

• The correct expanded form of $\left(\frac{1}{5}\right)^4$ is $\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5}$.
• To evaluate the expression, multiply the fractions together.
• The value of $\left(\frac{1}{5}\right)^4$ is $\frac{1}{625}$.
  Q&A: Understanding Exponents and Expanded Forms =====================================================

In our previous article, we explored the correct expanded form and value of $\left(\frac{1}{5}\right)^4$. In this article, we will continue to answer some of the most frequently asked questions related to exponents and expanded forms.

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

A: An exponent is a small number that is placed above and to the right of a number or expression, representing the number of times the base is multiplied by itself. A power, on the other hand, is the result of raising a base to a certain exponent. For example, in the expression $2^3$, the exponent 3 represents the number of times the base 2 is multiplied by itself, and the power is the result of this multiplication, which is 8.

Q: How do I evaluate an expression with a negative exponent?

A: When evaluating an expression with a negative exponent, you can rewrite the expression with a positive exponent by taking the reciprocal of the base. For example, in the expression $\frac{1}{2^{-3}}$, you can rewrite it as $2^3$ by taking the reciprocal of the base.

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

A: When multiplying exponents with the same base, you can add the exponents together. For example, in the expression $2^3 \cdot 2^4$, you can add the exponents together to get $2^{3+4} = 2^7$.

Q: How do I evaluate an expression with a zero exponent?

A: When evaluating an expression with a zero exponent, the result is always 1. For example, in the expression $2^0$, the result is 1.

Q: What is the rule for dividing exponents with the same base?

A: When dividing exponents with the same base, you can subtract the exponents. For example, in the expression $\frac{2^5}{2^3}$, you can subtract the exponents to get $2^{5-3} = 2^2$.

Q: How do I evaluate an expression with a fractional exponent?

A: When evaluating an expression with a fractional exponent, you can rewrite the expression as a root. For example, in the expression $\sqrt[3]{2^2}$, you can rewrite it as $2^{\frac{2}{3}}$.

Q: What is the rule for raising a power to a power?

A: When raising a power to a power, you can multiply the exponents together. For example, in the expression $(2^3)^4$, you can multiply the exponents together to get $2^{3 \cdot 4} = 2^{12}$.

Q: How do I evaluate an expression with a negative base and a positive exponent?

A: When evaluating an expression with a negative base and a positive exponent, you can rewrite the expression as a positive base raised to the same exponent. For example, in the expression $(-2)^3$, you can rewrite it as $(-2)^3 = -2 \cdot -2 \cdot -2 = -8$.

Q: What is the rule for evaluating an expression with a variable base and a variable exponent?

A: When evaluating an expression with a variable base and a variable exponent, you can use the rules of exponents to simplify the expression. For example, in the expression $x^m \cdot x^n$, you can add the exponents together to get $x^{m+n}$.

Conclusion

In conclusion, understanding exponents and expanded forms is a crucial concept in mathematics. By following the rules of exponents and using the correct techniques for evaluating expressions, you can simplify complex expressions and solve problems with ease. We hope that this Q&A article has been helpful in answering some of the most frequently asked questions related to exponents and expanded forms.