Evaluate Each Expression.1. $\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = \frac{1}{5}$2. $\frac{27^2}{27^{\frac{4}{3}}} = \, \square$

Introduction

Mathematical expressions are a fundamental part of mathematics, and evaluating them is a crucial skill for students and professionals alike. In this article, we will evaluate two mathematical expressions and provide a step-by-step guide on how to simplify them.

Expression 1: $\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = \frac{1}{5}$

Understanding the Expression

The given expression is $\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}$. To evaluate this expression, we need to understand the properties of exponents. The expression can be broken down into two parts: the base and the exponent.

• The base is $25^{-\frac{3}{2}}$.
• The exponent is $\frac{1}{3}$.

Simplifying the Expression

To simplify the expression, we need to apply the property of exponents that states $(a^m)^n = a^{mn}$. In this case, we have:

$$\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}} = 25^{-\frac{3}{2} \cdot \frac{1}{3}}$$

Now, we can simplify the exponent:

$$-\frac{3}{2} \cdot \frac{1}{3} = -\frac{1}{2}$$

So, the expression becomes:

$$25^{-\frac{1}{2}}$$

Evaluating the Expression

To evaluate the expression, we need to understand that $25^{-\frac{1}{2}}$ is equivalent to $\frac{1}{\sqrt{25}}$. Since $\sqrt{25} = 5$, we can simplify the expression as follows:

$$25^{-\frac{1}{2}} = \frac{1}{\sqrt{25}} = \frac{1}{5}$$

Therefore, the expression $\left(25^{-\frac{3}{2}}\right)^{\frac{1}{3}}$ is equal to $\frac{1}{5}$.

Expression 2: $\frac{27^2}{27^{\frac{4}{3}}} = \, \square$

Understanding the Expression

The given expression is $\frac{27^2}{27^{\frac{4}{3}}}$. To evaluate this expression, we need to understand the properties of exponents. The expression can be broken down into two parts: the numerator and the denominator.

• The numerator is $27^2$.
• The denominator is $27^{\frac{4}{3}}$.

Simplifying the Expression

To simplify the expression, we need to apply the property of exponents that states $\frac{a^m}{a^n} = a^{m-n}$. In this case, we have:

$$\frac{27^2}{27^{\frac{4}{3}}} = 27^{2 - \frac{4}{3}}$$

Now, we can simplify the exponent:

$$2 - \frac{4}{3} = \frac{6}{3} - \frac{4}{3} = \frac{2}{3}$$

So, the expression becomes:

$$27^{\frac{2}{3}}$$

Evaluating the Expression

To evaluate the expression, we need to understand that $27^{\frac{2}{3}}$ is equivalent to $\sqrt[3]{27^2}$. Since $\sqrt[3]{27^2} = \sqrt[3]{729}$, we can simplify the expression as follows:

$$27^{\frac{2}{3}} = \sqrt[3]{27^2} = \sqrt[3]{729} = 9$$

Therefore, the expression $\frac{27^2}{27^{\frac{4}{3}}}$ is equal to $9$.

Conclusion

$$$$$$$$

Introduction

In our previous article, we evaluated two mathematical expressions and provided a step-by-step guide on how to simplify them. In this article, we will answer some frequently asked questions (FAQs) related to evaluating mathematical expressions.

Q: What are the properties of exponents?

A: The properties of exponents are a set of rules that help us simplify expressions with exponents. Some of the key properties of exponents include:

• Product of Powers: $a^m \cdot a^n = a^{m+n}$
• Power of a Power: $(a^m)^n = a^{mn}$
• Power of a Product: $(ab)^m = a^m \cdot b^m$
• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$

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

A: To simplify an expression with a negative exponent, we can use the property of exponents that states $a^{-m} = \frac{1}{a^m}$. For example, if we have the expression $25^{-\frac{3}{2}}$, we can simplify it as follows:

$$25^{-\frac{3}{2}} = \frac{1}{25^{\frac{3}{2}}}$$

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

A: To simplify an expression with a fractional exponent, we can use the property of exponents that states $a^{\frac{m}{n}} = \sqrt[n]{a^m}$. For example, if we have the expression $27^{\frac{2}{3}}$, we can simplify it as follows:

$$27^{\frac{2}{3}} = \sqrt[3]{27^2}$$

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

A: A radical and an exponent are both used to represent repeated multiplication, but they are used in different ways. A radical is used to represent a root, while an exponent is used to represent repeated multiplication. For example, the expression $\sqrt{25}$ is equivalent to $25^{\frac{1}{2}}$.

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

A: To evaluate an expression with multiple exponents, we can use the property of exponents that states $(a^m)^n = a^{mn}$. For example, if we have the expression $(25^{-\frac{3}{2}})^{\frac{1}{3}}$, we can simplify it as follows:

$$(25^{-\frac{3}{2}})^{\frac{1}{3}} = 25^{-\frac{3}{2} \cdot \frac{1}{3}}$$

Q: What are some common mistakes to avoid when evaluating mathematical expressions?

A: Some common mistakes to avoid when evaluating mathematical expressions include:

• Not following the order of operations: Make sure to follow the order of operations (PEMDAS) when evaluating expressions.
• Not simplifying expressions: Make sure to simplify expressions before evaluating them.
• Not using the correct properties of exponents: Make sure to use the correct properties of exponents when simplifying expressions.

Conclusion

In this article, we answered some frequently asked questions related to evaluating mathematical expressions. We covered topics such as the properties of exponents, simplifying expressions with negative and fractional exponents, and evaluating expressions with multiple exponents. We hope that this article has provided a clear understanding of how to evaluate mathematical expressions and has helped students and professionals alike to simplify complex expressions.