Evaluate The Expression:${ \left{6 2+\left(8 2\right)^{3 / 2}\right}^3 }$

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Introduction

In mathematics, evaluating expressions is a crucial skill that involves simplifying complex mathematical expressions to obtain a final value. This skill is essential in various mathematical disciplines, including algebra, geometry, and calculus. In this article, we will evaluate the expression $\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3$ using mathematical principles and rules.

Understanding the Expression

The given expression is $\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3$. To evaluate this expression, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Evaluating the Expression

Let's start by evaluating the expression inside the parentheses:

$\left(8^2\right)^{3 / 2}$

Using the rule of exponents, we can rewrite this expression as:

$8^{2 \cdot \frac{3}{2}}$

Simplifying the exponent, we get:

$8^3$

Evaluating this expression, we get:

$8^3 = 512$

Now, let's evaluate the expression $6^2$:

$6^2 = 36$

Combining the Results

Now that we have evaluated the expressions inside the parentheses, we can combine the results:

$6^2 + \left(8^2\right)^{3 / 2} = 36 + 512$

Evaluating this expression, we get:

$36 + 512 = 548$

Raising to the Power of 3

Finally, we need to raise the result to the power of 3:

$\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3 = 548^3$

Evaluating this expression, we get:

$548^3 = 168, 665, 472$

Conclusion

In this article, we evaluated the expression $\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3$ using mathematical principles and rules. We followed the order of operations (PEMDAS) and evaluated the expressions inside the parentheses first. We then combined the results and raised the final result to the power of 3. The final value of the expression is $168, 665, 472$.

Frequently Asked Questions

• What is the order of operations (PEMDAS)? The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for:
  • Parentheses: Evaluate expressions inside parentheses first.
  • Exponents: Evaluate any exponential expressions next.
  • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
• How do I evaluate an expression with exponents? To evaluate an expression with exponents, you need to follow the rule of exponents, which states that:
  • $a^m \cdot a^n = a^{m+n}$
  • $\left(a^m\right)^n = a^{m \cdot n}$
• What is the difference between a variable and a constant? A variable is a symbol that represents a value that can change, while a constant is a value that remains the same.

Final Answer

The final answer to the expression $\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3$ is $168, 665, 472$.

Introduction

Evaluating mathematical expressions is a crucial skill that involves simplifying complex mathematical expressions to obtain a final value. In our previous article, we evaluated the expression $\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3$ using mathematical principles and rules. In this article, we will answer some frequently asked questions related to evaluating mathematical expressions.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictates the order in which mathematical operations should be performed. The acronym PEMDAS stands for: + Parentheses: Evaluate expressions inside parentheses first. + Exponents: Evaluate any exponential expressions next. + Multiplication and Division: Evaluate any multiplication and division operations from left to right. + Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate an expression with exponents?

A: To evaluate an expression with exponents, you need to follow the rule of exponents, which states that: + $a^m \cdot a^n = a^{m+n}$ + $\left(a^m\right)^n = a^{m \cdot n}$

Q: What is the difference between a variable and a constant?

A: A variable is a symbol that represents a value that can change, while a constant is a value that remains the same.

Q: How do I simplify a complex mathematical expression?

A: To simplify a complex mathematical expression, you need to follow the order of operations (PEMDAS) and evaluate the expressions inside the parentheses first. Then, you can combine the results and simplify the expression further.

Q: What is the importance of evaluating mathematical expressions?

A: Evaluating mathematical expressions is important because it helps you to: + Simplify complex mathematical expressions + Solve mathematical problems + Understand mathematical concepts + Develop problem-solving skills

Q: How do I evaluate an expression with fractions?

A: To evaluate an expression with fractions, you need to follow the rules of fractions, which state that: + $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ + $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers. An irrational expression is an expression that cannot be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers.

Q: How do I evaluate an expression with absolute value?

A: To evaluate an expression with absolute value, you need to follow the rules of absolute value, which state that: + $|a| = a$ if $a \geq 0$ + $|a| = -a$ if $a < 0$

Conclusion

Evaluating mathematical expressions is a crucial skill that involves simplifying complex mathematical expressions to obtain a final value. In this article, we answered some frequently asked questions related to evaluating mathematical expressions. We hope that this article has helped you to understand the importance of evaluating mathematical expressions and how to evaluate them.

Final Answer

The final answer to the expression $\left\{6^2+\left(8^2\right)^{3 / 2}\right\}^3$ is $168, 665, 472$.