Evaluate Each Expression.1. $\left(25 {-\frac{3}{2}}\right) {\frac{1}{3}} = \frac{1}{5}$2. 27 2 27 1 3 = \frac{27^2}{27^{\frac{1}{3}}} = 2 7 3 1 ​ 2 7 2 ​ =

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 and how to simplify them.

Simplifying the Expression

We can start by simplifying the expression inside the parentheses. We have $25^{-\frac{3}{2}}$, which can be rewritten as $\frac{1}{25^{\frac{3}{2}}}$. Now, we can simplify the exponent by using the property of exponents that states $\left(a^m\right)^n = a^{mn}$.

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

Applying the Property of Exponents

Now, we can apply the property of exponents to simplify the expression. We have $\left(\frac{1}{25^{\frac{3}{2}}}\right)^{\frac{1}{3}}$, which can be rewritten as $\frac{1}{\left(25^{\frac{3}{2}}\right)^{\frac{1}{3}}}$. Using the property of exponents, we can simplify this expression to $\frac{1}{25^{\frac{1}{2}}}$.

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

Simplifying the Expression Further

Now, we can simplify the expression further by rewriting $25^{\frac{1}{2}}$ as $\sqrt{25}$. We know that $\sqrt{25} = 5$, so we can substitute this value into the expression.

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

Conclusion

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

**Expression 2: $\frac{27^2}{27^{\frac{1}{3}}} =$

Understanding the Expression

The given expression is $\frac{27^2}{27^{\frac{1}{3}}}$. To evaluate this expression, we need to understand the properties of exponents and how to simplify them.

Simplifying the Expression

We can start by simplifying the numerator and denominator separately. We have $27^2$ in the numerator, which can be rewritten as $27 \times 27$. We also have $27^{\frac{1}{3}}$ in the denominator, which can be rewritten as $\sqrt[3]{27}$.

$\frac{27^2}{27^{\frac{1}{3}}} = \frac{27 \times 27}{\sqrt[3]{27}}$

Applying the Property of Exponents

Now, we can apply the property of exponents to simplify the expression. We have $\frac{27 \times 27}{\sqrt[3]{27}}$, which can be rewritten as $\frac{27^2}{27^{\frac{1}{3}}}$. Using the property of exponents, we can simplify this expression to $27^{\frac{2}{3}}$.

$\frac{27^2}{27^{\frac{1}{3}}} = 27^{\frac{2}{3}}$

Simplifying the Expression Further

Now, we can simplify the expression further by rewriting $27^{\frac{2}{3}}$ as $\left(\sqrt[3]{27}\right)^2$. We know that $\sqrt[3]{27} = 3$, so we can substitute this value into the expression.

$27^{\frac{2}{3}} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$

Conclusion

Therefore, the value of the expression $\frac{27^2}{27^{\frac{1}{3}}}$ is $9$.

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: $a^m \times a^n = a^{m+n}$
• Power of a Power: $\left(a^m\right)^n = a^{mn}$
• Power of a Product: $\left(ab\right)^n = a^n \times b^n$
• Power of a Quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

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

A: To simplify an expression with a negative exponent, we can rewrite it as a fraction with a positive exponent. For example, $a^{-m}$ can be rewritten as $\frac{1}{a^m}$.

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

A: To simplify an expression with a fractional exponent, we can rewrite it as a product of a number and a root. For example, $a^{\frac{m}{n}}$ can be rewritten as $\sqrt[n]{a^m}$.

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.

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

A: To evaluate an expression with multiple exponents, we need to follow the order of operations (PEMDAS):

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponents.
3. Multiply any numbers.
4. Divide any numbers.

Q: How do I simplify an expression with a variable in the exponent?

A: To simplify an expression with a variable in the exponent, we need to follow the rules of exponents. For example, if we have $a^{2x}$, we can rewrite it as $(a^x)^2$.

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression with a single variable and a degree of 1, while a quadratic expression is an expression with a single variable and a degree of 2.

Q: How do I evaluate an expression with a variable in the denominator?

A: To evaluate an expression with a variable in the denominator, we need to follow the rules of fractions. For example, if we have $\frac{1}{x}$, we can rewrite it as $x^{-1}$.

Conclusion

In this article, we answered some frequently asked questions related to evaluating mathematical expressions. We hope that this article has provided a clear understanding of how to evaluate mathematical expressions and has helped readers to develop their problem-solving skills.

Additional Resources

• Mathematical Expressions: A comprehensive guide to mathematical expressions, including rules, properties, and examples.
• Exponents: A detailed explanation of exponents, including rules, properties, and examples.
• Radicals: A comprehensive guide to radicals, including rules, properties, and examples.
• Order of Operations: A detailed explanation of the order of operations, including PEMDAS and examples.

Practice Problems

• Evaluate the expression $2^3 \times 2^4$.
• Simplify the expression $\frac{1}{x^2} \times x^3$.
• Evaluate the expression $3^2 + 3^3$.
• Simplify the expression $\frac{1}{2^3} \times 2^4$.

We hope that this article has provided a clear understanding of how to evaluate mathematical expressions and has helped readers to develop their problem-solving skills.