Simplify The Expression: { ( -8 B A^3 )^2$}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When dealing with algebraic expressions, we often encounter exponents and powers that need to be simplified. In this article, we will focus on simplifying the expression {( -8 b a^3 )^2$}$, which involves exponentiation and algebraic manipulation.

Understanding Exponents and Powers

Before we dive into simplifying the given expression, let's review the basics of exponents and powers. An exponent is a small number that is written to the upper right of a number or a variable, indicating how many times the base is multiplied by itself. For example, in the expression ${a^3}$, the exponent 3 indicates that the base ${a}$ is multiplied by itself 3 times: ${a \times a \times a}$.

Simplifying the Expression

To simplify the expression {( -8 b a^3 )^2$}$, we need to apply the exponentiation rule, which states that ${(ab)^n = a^n \times b^n}$. In this case, we have ${(-8ba^3)^2}$, which can be expanded as:

${(-8ba^3)^2 = (-8)^2 \times (ba^3)^2}$

Expanding the Terms

Now, let's expand the terms inside the parentheses:

${(-8)^2 = 64}$

${(ba^3)^2 = b^2 \times (a^3)^2}$

${(a^3)^2 = a^6}$

Combining the Terms

Now that we have expanded the terms, we can combine them to simplify the expression:

${64 \times b^2 \times a^6}$

Simplifying the Expression Further

We can simplify the expression further by combining the constants and the variables:

${64b^2a^6}$

Conclusion

In this article, we simplified the expression {( -8 b a^3 )^2$}$ using exponentiation and algebraic manipulation. We reviewed the basics of exponents and powers, expanded the terms inside the parentheses, and combined them to simplify the expression. The final simplified expression is ${64b^2a^6}$.

Frequently Asked Questions

• What is the exponentiation rule? The exponentiation rule states that ${(ab)^n = a^n \times b^n}$.
• How do I simplify an expression with exponents? To simplify an expression with exponents, you need to apply the exponentiation rule and expand the terms inside the parentheses.
• What is the difference between an exponent and a power? An exponent is a small number that is written to the upper right of a number or a variable, indicating how many times the base is multiplied by itself. A power is the result of raising a base to a certain exponent.

Final Thoughts

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently and accurately. By understanding exponents and powers, we can simplify complex expressions and arrive at the correct solution. In this article, we simplified the expression {( -8 b a^3 )^2$}$ using exponentiation and algebraic manipulation. We hope that this article has provided you with a better understanding of how to simplify expressions with exponents.

Additional Resources

• Khan Academy: Exponents and Powers
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponentiation and Algebraic Manipulation

Related Articles

• Simplifying Expressions with Fractions
• Solving Equations with Exponents
• Understanding Algebraic Manipulation
  

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Introduction

In our previous article, we simplified the expression {( -8 b a^3 )^2$}$ using exponentiation and algebraic manipulation. In this article, we will answer some frequently asked questions related to simplifying expressions with exponents.

Q&A

Q: What is the exponentiation rule?

A: The exponentiation rule states that ${(ab)^n = a^n \times b^n}$. This rule helps us simplify expressions with exponents by expanding the terms inside the parentheses.

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, you need to apply the exponentiation rule and expand the terms inside the parentheses. You can also use the power rule, which states that ${(a^m)^n = a^{mn}}$.

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

A: An exponent is a small number that is written to the upper right of a number or a variable, indicating how many times the base is multiplied by itself. A power is the result of raising a base to a certain exponent.

Q: How do I handle negative exponents?

A: When dealing with negative exponents, you can rewrite them as fractions. For example, ${a^{-n} = \frac{1}{a^n}}$.

Q: Can I simplify expressions with multiple exponents?

A: Yes, you can simplify expressions with multiple exponents by applying the exponentiation rule and expanding the terms inside the parentheses. You can also use the power rule to simplify expressions with multiple exponents.

Q: How do I simplify expressions with variables in the exponent?

A: When dealing with variables in the exponent, you can use the power rule to simplify the expression. For example, ${(a^m)^n = a^{mn}}$.

Q: Can I simplify expressions with fractions in the exponent?

A: Yes, you can simplify expressions with fractions in the exponent by applying the exponentiation rule and expanding the terms inside the parentheses.

Q: How do I handle complex expressions with exponents?

A: When dealing with complex expressions with exponents, you can break them down into smaller parts and simplify each part separately. You can also use the power rule and the exponentiation rule to simplify the expression.

Examples

Example 1: Simplifying an expression with a single exponent

Simplify the expression ${(2x^3)^2}$.

Solution:

${(2x^3)^2 = (2)^2 \times (x^3)^2}$

${= 4 \times x^6}$

Example 2: Simplifying an expression with multiple exponents

Simplify the expression ${(2x^3y^2)^4}$.

Solution:

${(2x^3y^2)^4 = (2)^4 \times (x^3)^4 \times (y^2)^4}$

${= 16 \times x^{12} \times y^8}$

Example 3: Simplifying an expression with a variable in the exponent

Simplify the expression ${(x^m)^n}$.

Solution:

${(x^m)^n = x^{mn}}$

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with exponents. We also provided examples to illustrate how to simplify expressions with single exponents, multiple exponents, and variables in the exponent. By understanding the exponentiation rule and the power rule, you can simplify complex expressions with exponents and arrive at the correct solution.

Frequently Asked Questions

• What is the exponentiation rule?
• How do I simplify an expression with exponents?
• What is the difference between an exponent and a power?
• How do I handle negative exponents?
• Can I simplify expressions with multiple exponents?
• How do I simplify expressions with variables in the exponent?
• Can I simplify expressions with fractions in the exponent?
• How do I handle complex expressions with exponents?

Additional Resources

• Khan Academy: Exponents and Powers
• Mathway: Simplifying Expressions with Exponents
• Wolfram Alpha: Exponentiation and Algebraic Manipulation

Related Articles

• Simplifying Expressions with Fractions
• Solving Equations with Exponents
• Understanding Algebraic Manipulation