Which Of The Following Is Equal To The Expression \[$\left(10^2 \cdot 3\right)^{\frac{1}{2}}\$\]?A. \[$\sqrt{3}\$\]B. \[$10 \sqrt{3}\$\]C. \[$\sqrt{30}\$\]D. \[$\sqrt{3,000}\$\]

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Introduction

Exponential expressions can be complex and challenging to simplify, but with the right approach, they can be broken down into manageable parts. In this article, we will explore how to simplify the expression {\left(10^2 \cdot 3\right)^{\frac{1}{2}}$}$ and determine which of the given options is equal to it.

Understanding Exponents

Before we dive into the simplification process, let's review the basics of exponents. An exponent is a small number that is written to the upper right of a number or expression. It represents the power to which the base number or expression is raised. For example, in the expression ${2^3\$}, the exponent ${3\$} indicates that the base number ${2\$} is raised to the power of ${3\$}.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the given expression {\left(10^2 \cdot 3\right)^{\frac{1}{2}}$}$. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expression inside the parentheses first. In this case, we have ${10^2 \cdot 3\$}.
  2. Exponents: Evaluate the exponent ${10^2\$} first. This is equal to ${100\$}.
  3. Multiplication: Multiply the result by ${3\$}. This gives us ${300\$}.
  4. Root: Finally, take the square root of the result. This is equal to {\sqrt{300}$}$.

Simplifying the Square Root

Now that we have simplified the expression inside the parentheses, let's simplify the square root. To simplify the square root of ${300\$}, we need to find the largest perfect square that divides ${300\$}. In this case, the largest perfect square that divides ${300\$} is ${100\$}. Therefore, we can rewrite the square root as:

{\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10 \sqrt{3}$}$

Conclusion

In conclusion, the expression {\left(10^2 \cdot 3\right)^{\frac{1}{2}}$}$ simplifies to ${10 \sqrt{3}\$}. This is equal to option B.

Comparison with Other Options

Let's compare our result with the other options:

  • Option A: {\sqrt{3}$}$ is not equal to our result.
  • Option C: {\sqrt{30}$}$ is not equal to our result.
  • Option D: {\sqrt{3,000}$}$ is not equal to our result.

Final Answer

The final answer is option B: ${10 \sqrt{3}\$}.

Additional Tips and Tricks

Here are some additional tips and tricks to help you simplify exponential expressions:

  • Use the order of operations: Always follow the order of operations (PEMDAS) when simplifying exponential expressions.
  • Simplify inside the parentheses: Simplify the expression inside the parentheses before evaluating the exponent.
  • Use the properties of exponents: Use the properties of exponents to simplify the expression. For example, you can use the property {a^m \cdot a^n = a^{m+n}$}$ to simplify the expression.
  • Simplify the square root: Simplify the square root by finding the largest perfect square that divides the number.

Introduction

In our previous article, we explored how to simplify the expression {\left(10^2 \cdot 3\right)^{\frac{1}{2}}$}$ and determined that it is equal to ${10 \sqrt{3}\$}. In this article, we will answer some frequently asked questions about simplifying exponential expressions.

Q&A

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

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

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

A: To simplify an expression with multiple exponents, you need to follow the order of operations. First, evaluate the exponents from left to right. Then, simplify the expression inside the parentheses. Finally, simplify the expression using the properties of exponents.

Q: What is the property of exponents that states {a^m \cdot a^n = a^{m+n}$]?

A: This property is called the product of powers property. It states that when you multiply two powers with the same base, you can add the exponents.

Q: How do I simplify a square root?

A: To simplify a square root, you need to find the largest perfect square that divides the number. For example, to simplify [\sqrt{300}\$}, you need to find the largest perfect square that divides ${300\$}. In this case, the largest perfect square that divides ${300\$} is ${100\$}. Therefore, you can rewrite the square root as:

{\sqrt{300} = \sqrt{100 \cdot 3} = \sqrt{100} \cdot \sqrt{3} = 10 \sqrt{3}$}$

Q: What is the difference between a square root and a cube root?

A: A square root is a root that is raised to the power of {\frac{1}{2}$}$. A cube root is a root that is raised to the power of {\frac{1}{3}$}$. For example, {\sqrt{4}$] is equal to [2\$}, while {\sqrt[3]{8}$] is equal to [2\$}.

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

A: To simplify an expression with a negative exponent, you need to follow the rule that states {a^{-m} = \frac{1}{a^m}$}$. For example, to simplify ${2^{-3}\$}, you need to rewrite it as:

${2^{-3} = \frac{1}{2^3} = \frac{1}{8}\$}

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains a base raised to a power. A polynomial expression is an expression that contains variables raised to various powers, but no exponents. For example, ${2^3 + 3x^2\$} is an exponential expression, while ${3x^2 + 2x + 1\$} is a polynomial expression.

Conclusion

In conclusion, simplifying exponential expressions can be challenging, but with the right approach, it can be done with ease. By following the order of operations, using the properties of exponents, and simplifying the square root, you can simplify even the most complex exponential expressions. We hope that this Q&A guide has been helpful in answering your questions about simplifying exponential expressions.