Algebraic Expressions Calculator

Oct 13, 2025 by miodeya.com 149 views

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with algebraic expressions, we often encounter exponents and variables that need to be manipulated to simplify the expression. In this article, we will focus on simplifying the given expression $\left(-3 a^3\right)(-3 a)^{-3}$ using the properties of exponents and variables.

Understanding the Expression

The given expression is a product of two terms: $\left(-3 a^3\right)$ and $\left(-3 a\right)^{-3}$ . To simplify this expression, we need to understand the properties of exponents and how they interact with variables. The first term, $\left(-3 a^3\right)$ , is a product of a constant and a variable raised to the power of 3. The second term, $\left(-3 a\right)^{-3}$ , is a product of a constant and a variable raised to the power of -3.

Properties of Exponents

Before we simplify the expression, let's review the properties of exponents that we will use:

Product of Powers: When multiplying two powers with the same base, we add the exponents. For example, $a^m \cdot a^n = a^{m+n}$ .
Power of a Power: When raising a power to another power, we multiply the exponents. For example, $\left(a^m\right)^n = a^{m \cdot n}$ .
Negative Exponent: A negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$ .

Simplifying the Expression

Now that we have reviewed the properties of exponents, let's simplify the given expression:

$\left(-3 a^3\right)(-3 a)^{-3}$

Using the property of negative exponents, we can rewrite the second term as:

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

Now, we can simplify the expression by multiplying the two terms:

$\left(-3 a^3\right)\left(\frac{1}{\left(-3 a\right)^3}\right)$

Using the property of product of powers, we can simplify the expression further:

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

Canceling Out Common Factors

Now that we have simplified the expression, we can cancel out common factors in the numerator and denominator:

$\frac{\left(-3 a^3\right)}{\left(-3 a\right)^3} = \frac{-3 a^3}{-3^3 a^3}$

Using the property of negative exponents, we can rewrite the denominator as:

$\frac{-3 a^3}{-3^3 a^3} = \frac{-3 a^3}{-27 a^3}$

Final Simplification

Now that we have canceled out common factors, we can simplify the expression further:

$\frac{-3 a^3}{-27 a^3} = \frac{1}{9}$

Conclusion

In this article, we simplified the expression $\left(-3 a^3\right)(-3 a)^{-3}$ using the properties of exponents and variables. We reviewed the properties of exponents, including the product of powers, power of a power, and negative exponent. We then applied these properties to simplify the expression, canceling out common factors and arriving at the final simplified expression: $\frac{1}{9}$ . This example demonstrates the importance of understanding the properties of exponents and how they interact with variables in algebraic expressions.

Frequently Asked Questions

What is the property of negative exponents? A negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$ .
How do we simplify expressions with exponents? We can simplify expressions with exponents by applying the properties of exponents, including the product of powers, power of a power, and negative exponent.
What is the final simplified expression? The final simplified expression is $\frac{1}{9}$ .

Introduction

In our previous article, we simplified the expression $\left(-3 a^3\right)(-3 a)^{-3}$ using the properties of exponents and variables. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents and variables.

Q&A

Q: What is the property of negative exponents?

A: A negative exponent indicates that we need to take the reciprocal of the base raised to the positive exponent. For example, $a^{-m} = \frac{1}{a^m}$ .

Q: How do we simplify expressions with exponents?

A: We can simplify expressions with exponents by applying the properties of exponents, including the product of powers, power of a power, and negative exponent.

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{1}{9}$ .

Q: Can we simplify expressions with variables in the exponent?

A: Yes, we can simplify expressions with variables in the exponent by applying the properties of exponents. For example, $\left(a^m\right)\left(a^n\right) = a^{m+n}$ .

Q: How do we handle negative exponents in the denominator?

A: When a negative exponent is in the denominator, we can rewrite it as a positive exponent in the numerator. For example, $\frac{1}{a^{-m}} = a^m$ .

Q: Can we simplify expressions with multiple variables?

A: Yes, we can simplify expressions with multiple variables by applying the properties of exponents. For example, $\left(a^m\right)\left(b^n\right) = a^m b^n$ .

Q: How do we handle exponents with different bases?

A: When exponents have different bases, we can simplify the expression by applying the properties of exponents. For example, $\left(a^m\right)\left(b^n\right) = a^m b^n$ .

Q: Can we simplify expressions with fractional exponents?

A: Yes, we can simplify expressions with fractional exponents by applying the properties of exponents. For example, $\left(a^{\frac{m}{n}}\right) = \sqrt[n]{a^m}$ .

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with exponents and variables. We reviewed the properties of exponents, including the product of powers, power of a power, and negative exponent. We also provided examples of how to simplify expressions with multiple variables, different bases, and fractional exponents.

Additional Resources

Mathematics Glossary: This article provides a comprehensive glossary of mathematical terms and concepts.
Algebraic Expressions Calculator: This online calculator can help you simplify algebraic expressions with exponents and variables.
Mathematics Tutorials: This website provides a collection of mathematics tutorials and examples on various topics, including algebra and calculus.