Write $\frac{\sqrt{8^3}}{\sqrt{4}}$ As A Single Radical Using The Smallest Possible Root.

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Introduction

Radicals are an essential part of mathematics, and they play a crucial role in algebra and geometry. In this article, we will focus on simplifying a given expression involving radicals. The expression we will be working with is 834\frac{\sqrt{8^3}}{\sqrt{4}}. Our goal is to rewrite this expression as a single radical using the smallest possible root.

Understanding Radicals

Radicals are mathematical expressions that involve the use of roots. The most common radical is the square root, denoted by x\sqrt{x}. However, there are other types of radicals, such as cube roots, fourth roots, and so on. In this article, we will be dealing with square roots and cube roots.

Simplifying the Expression

To simplify the given expression, we need to start by evaluating the numerator and denominator separately. The numerator is 83\sqrt{8^3}, and the denominator is 4\sqrt{4}.

Evaluating the Numerator

The numerator is 83\sqrt{8^3}. To evaluate this expression, we need to start by simplifying the expression inside the square root. We can do this by using the property of exponents that states am⋅an=am+na^m \cdot a^n = a^{m+n}. In this case, we have 83=8⋅8⋅8=23⋅23⋅23=298^3 = 8 \cdot 8 \cdot 8 = 2^3 \cdot 2^3 \cdot 2^3 = 2^9.

import math

# Define the variables
numerator = math.sqrt(8**3)

# Simplify the expression inside the square root
simplified_numerator = math.sqrt(2**9)

Evaluating the Denominator

The denominator is 4\sqrt{4}. To evaluate this expression, we can simply take the square root of 4, which is equal to 2.

# Define the variable
denominator = math.sqrt(4)

# Evaluate the denominator
evaluated_denominator = 2

Combining the Results

Now that we have evaluated the numerator and denominator, we can combine the results to simplify the given expression. We can do this by dividing the numerator by the denominator.

# Combine the results
result = simplified_numerator / evaluated_denominator

Simplifying the Result

The result we obtained is 292\frac{\sqrt{2^9}}{2}. To simplify this expression further, we can use the property of radicals that states a2=a\sqrt{a^2} = a. In this case, we have 29=(24)2⋅2=222=42\sqrt{2^9} = \sqrt{(2^4)^2 \cdot 2} = 2^2 \sqrt{2} = 4 \sqrt{2}.

# Simplify the result
simplified_result = 4 * math.sqrt(2)

Conclusion

In this article, we have simplified the given expression 834\frac{\sqrt{8^3}}{\sqrt{4}} as a single radical using the smallest possible root. We started by evaluating the numerator and denominator separately, and then combined the results to simplify the expression. Finally, we simplified the result further using the properties of radicals. The final answer is 42\boxed{4 \sqrt{2}}.

Final Answer

The final answer is 42\boxed{4 \sqrt{2}}.

References

Tags

  • radicals
  • square roots
  • cube roots
  • simplifying expressions
  • mathematical properties
  • algebra
  • geometry

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Introduction

In our previous article, we discussed how to simplify the expression 834\frac{\sqrt{8^3}}{\sqrt{4}} as a single radical using the smallest possible root. In this article, we will answer some frequently asked questions related to simplifying radicals and expressions.

Q&A

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

A: A radical is a mathematical expression that involves the use of roots, while an exponent is a mathematical expression that involves the use of powers. For example, x\sqrt{x} is a radical, while x2x^2 is an exponent.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to start by evaluating the expression inside the radical. If the expression can be simplified further, you can use the properties of radicals to simplify it. For example, 83=(23)3=232=82\sqrt{8^3} = \sqrt{(2^3)^3} = 2^3 \sqrt{2} = 8 \sqrt{2}.

Q: What is the property of radicals that states a2=a\sqrt{a^2} = a?

A: This property is known as the "property of radicals" or "the square root of a square is the number itself". It states that if aa is a positive number, then a2=a\sqrt{a^2} = a. For example, 42=16=4\sqrt{4^2} = \sqrt{16} = 4.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to start by evaluating the expression inside the radical. If the expression can be simplified further, you can use the properties of radicals to simplify it. For example, x2=x\sqrt{x^2} = x if xx is a positive number.

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

A: A rational expression is a mathematical expression that involves the use of fractions, while a radical expression is a mathematical expression that involves the use of roots. For example, xy\frac{x}{y} is a rational expression, while x\sqrt{x} is a radical expression.

Q: How do I simplify a rational expression with a radical?

A: To simplify a rational expression with a radical, you need to start by simplifying the radical expression. If the radical expression can be simplified further, you can use the properties of radicals to simplify it. For example, xy=x12y\frac{\sqrt{x}}{y} = \frac{x^{\frac{1}{2}}}{y}.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying radicals and expressions. We have discussed the properties of radicals, how to simplify radical expressions, and how to simplify rational expressions with radicals. We hope that this article has been helpful in clarifying any confusion you may have had about simplifying radicals and expressions.

Final Answer

The final answer is that simplifying radicals and expressions is an important part of mathematics, and it requires a good understanding of the properties of radicals and how to apply them.

References

Tags

  • radicals
  • square roots
  • cube roots
  • simplifying expressions
  • mathematical properties
  • algebra
  • geometry
  • rational expressions
  • variable expressions