Evaluate The Expression:$\[ \left(\frac{1}{125}\right)^{2 / 3} \\]

Introduction

In mathematics, exponents are a fundamental concept used to represent repeated multiplication of a number. When evaluating expressions with exponents, it's essential to understand the rules and properties of exponents to simplify and solve complex expressions. In this article, we will focus on evaluating the expression $\left(\frac{1}{125}\right)^{2 / 3}$, exploring the properties of exponents, and providing a step-by-step guide to simplify the expression.

Understanding Exponents

Exponents are a shorthand way of representing repeated multiplication of a number. For example, $2^3$ can be read as "2 to the power of 3" or "2 cubed," which is equivalent to $2 \times 2 \times 2 = 8$. Exponents can also be negative, fractional, or even complex numbers.

Properties of Exponents

When working with exponents, it's crucial to understand the following properties:

• Product of Powers: When multiplying two numbers with the same base, add the exponents. For example, $a^m \times a^n = a^{m+n}$.
• Power of a Power: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.
• Zero Exponent: Any non-zero number raised to the power of 0 is equal to 1. For example, $a^0 = 1$.
• Negative Exponent: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

Evaluating the Expression

Now that we have a solid understanding of exponents and their properties, let's focus on evaluating the expression $\left(\frac{1}{125}\right)^{2 / 3}$.

Step 1: Simplify the Fraction

The first step is to simplify the fraction $\frac{1}{125}$. We can rewrite 125 as $5^3$, so the fraction becomes $\frac{1}{5^3}$.

Step 2: Apply the Power of a Power Property

Using the power of a power property, we can rewrite the expression as $\left(\frac{1}{5^3}\right)^{2 / 3}$. Now, we can apply the property by multiplying the exponents: $\frac{1}{5^{3 \times (2/3)}}$.

Step 3: Simplify the Exponent

Simplifying the exponent, we get $\frac{1}{5^2}$.

Step 4: Evaluate the Expression

Finally, we can evaluate the expression by simplifying the fraction: $\frac{1}{25}$.

Conclusion

In this article, we evaluated the expression $\left(\frac{1}{125}\right)^{2 / 3}$ using the properties of exponents. We simplified the fraction, applied the power of a power property, and finally evaluated the expression to get the result $\frac{1}{25}$. By understanding the properties of exponents and applying them step-by-step, we can simplify complex expressions and solve problems with ease.

Additional Examples

To further reinforce your understanding of exponents, let's consider a few more examples:

• $\left(\frac{1}{8}\right)^{3 / 2}$
• $\left(\frac{1}{27}\right)^{2 / 3}$
• $\left(\frac{1}{64}\right)^{5 / 2}$

Try simplifying these expressions using the properties of exponents and see if you get the same results as we did in this article.

Final Thoughts

Introduction

In our previous article, we explored the concept of evaluating expressions with exponents, focusing on the expression $\left(\frac{1}{125}\right)^{2 / 3}$. We applied the properties of exponents, including the product of powers, power of a power, zero exponent, and negative exponent properties, to simplify the expression and arrive at the final result. In this article, we'll take a Q&A approach to further reinforce your understanding of evaluating expressions with exponents.

Q&A: Evaluating Expressions with Exponents

Q: What is the rule for simplifying a fraction with an exponent?

A: To simplify a fraction with an exponent, you can rewrite the fraction as a product of the numerator and the denominator raised to the power of the exponent. For example, $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$.

Q: How do you handle negative exponents?

A: A negative exponent can be rewritten as a positive exponent by taking the reciprocal of the base. For example, $a^{-m} = \frac{1}{a^m}$.

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

A: When multiplying two numbers with the same base, add the exponents. For example, $a^m \times a^n = a^{m+n}$.

Q: How do you handle zero exponents?

A: Any non-zero number raised to the power of 0 is equal to 1. For example, $a^0 = 1$.

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

A: When raising a power to another power, multiply the exponents. For example, $(a^m)^n = a^{m \times n}$.

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

A: To simplify an expression with multiple exponents, apply the properties of exponents in the correct order. For example, $\left(\frac{1}{5^3}\right)^{2 / 3} = \frac{1}{5^{3 \times (2/3)}} = \frac{1}{5^2}$.

Q: What is the final result of the expression $\left(\frac{1}{8}\right)^{3 / 2}$?

A: To simplify this expression, we can rewrite 8 as $2^3$, so the expression becomes $\left(\frac{1}{2^3}\right)^{3 / 2}$. Applying the power of a power property, we get $\frac{1}{2^{3 \times (3/2)}} = \frac{1}{2^4.5}$. Since we can't have a fraction as an exponent, we can rewrite this as $\frac{1}{2^4} \times \frac{1}{\sqrt{2}} = \frac{1}{16} \times \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{16}$.

Q: What is the final result of the expression $\left(\frac{1}{27}\right)^{2 / 3}$?

A: To simplify this expression, we can rewrite 27 as $3^3$, so the expression becomes $\left(\frac{1}{3^3}\right)^{2 / 3}$. Applying the power of a power property, we get $\frac{1}{3^{3 \times (2/3)}} = \frac{1}{3^2} = \frac{1}{9}$.

Q: What is the final result of the expression $\left(\frac{1}{64}\right)^{5 / 2}$?

A: To simplify this expression, we can rewrite 64 as $2^6$, so the expression becomes $\left(\frac{1}{2^6}\right)^{5 / 2}$. Applying the power of a power property, we get $\frac{1}{2^{6 \times (5/2)}} = \frac{1}{2^{15}} = \frac{1}{32768}$.

Conclusion

In this Q&A article, we've explored the concept of evaluating expressions with exponents, focusing on the properties of exponents and how to apply them to simplify complex expressions. We've also provided examples and solutions to common questions, reinforcing your understanding of this fundamental concept in mathematics. By mastering the properties of exponents, you'll be able to tackle even the most challenging problems with confidence and ease.

Additional Resources

For further practice and reinforcement, we recommend exploring the following resources:

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

Final Thoughts

Evaluating expressions with exponents requires a solid understanding of the properties of exponents. By applying the product of powers, power of a power, zero exponent, and negative exponent properties, you can simplify complex expressions and solve problems with ease. Remember to always simplify fractions, apply the power of a power property, and evaluate the expression to get the final result. With practice and patience, you'll become proficient in evaluating expressions with exponents and tackle even the most challenging problems with confidence.