Simplify The Expression: 8 4 6 \frac{8}{4 \sqrt{6}} 4 6 ​ 8 ​

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Introduction

Simplifying expressions is a crucial skill in mathematics, and it's essential to understand how to simplify various types of expressions, including those involving fractions and square roots. In this article, we will focus on simplifying the expression 846\frac{8}{4 \sqrt{6}}. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

The given expression is 846\frac{8}{4 \sqrt{6}}. This expression involves a fraction with a square root in the denominator. To simplify this expression, we need to understand the properties of fractions and square roots.

Properties of Fractions

A fraction is a way of expressing a part of a whole. It consists of a numerator (the number on top) and a denominator (the number on the bottom). In this expression, the numerator is 8 and the denominator is 464 \sqrt{6}.

Properties of Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. In this expression, the square root is 6\sqrt{6}.

Simplifying the Expression

To simplify the expression 846\frac{8}{4 \sqrt{6}}, we need to follow a series of steps. These steps involve simplifying the fraction and the square root separately.

Step 1: Simplify the Fraction

The fraction 84\frac{8}{4} can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4.

84=8÷44÷4=21\frac{8}{4} = \frac{8 \div 4}{4 \div 4} = \frac{2}{1}

So, the simplified fraction is 21\frac{2}{1}.

Step 2: Simplify the Square Root

The square root 6\sqrt{6} cannot be simplified further because 6 is a prime number and cannot be factored into the product of two numbers.

Step 3: Combine the Simplified Fraction and Square Root

Now that we have simplified the fraction and the square root separately, we can combine them to get the final simplified expression.

846=21÷6\frac{8}{4 \sqrt{6}} = \frac{2}{1} \div \sqrt{6}

To divide by a square root, we can multiply by its reciprocal.

21÷6=21×16\frac{2}{1} \div \sqrt{6} = \frac{2}{1} \times \frac{1}{\sqrt{6}}

Now, we can simplify the expression by multiplying the numerators and the denominators.

21×16=2×11×6=26\frac{2}{1} \times \frac{1}{\sqrt{6}} = \frac{2 \times 1}{1 \times \sqrt{6}} = \frac{2}{\sqrt{6}}

Step 4: Rationalize the Denominator

The expression 26\frac{2}{\sqrt{6}} has an irrational denominator. To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator.

The conjugate of 6\sqrt{6} is 6\sqrt{6} itself.

26×66=2×66×6\frac{2}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{2 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}

Now, we can simplify the expression by multiplying the numerators and the denominators.

2×66×6=266\frac{2 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}} = \frac{2\sqrt{6}}{6}

Step 5: Final Simplification

The expression 266\frac{2\sqrt{6}}{6} can be simplified further by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

266=26÷26÷2=63\frac{2\sqrt{6}}{6} = \frac{2\sqrt{6} \div 2}{6 \div 2} = \frac{\sqrt{6}}{3}

So, the final simplified expression is 63\frac{\sqrt{6}}{3}.

Conclusion

Simplifying the expression 846\frac{8}{4 \sqrt{6}} involves a series of steps, including simplifying the fraction and the square root separately, combining them, and rationalizing the denominator. By following these steps, we can simplify the expression to its final form, which is 63\frac{\sqrt{6}}{3}.

Frequently Asked Questions

Q: What is the simplified form of the expression 846\frac{8}{4 \sqrt{6}}?

A: The simplified form of the expression 846\frac{8}{4 \sqrt{6}} is 63\frac{\sqrt{6}}{3}.

Q: How do I simplify a fraction with a square root in the denominator?

A: To simplify a fraction with a square root in the denominator, you need to follow a series of steps, including simplifying the fraction and the square root separately, combining them, and rationalizing the denominator.

Q: What is the conjugate of a square root?

A: The conjugate of a square root is the same square root itself.

References

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Q&A: Simplifying Expressions with Square Roots

Q: What is the simplified form of the expression 846\frac{8}{4 \sqrt{6}}?

A: The simplified form of the expression 846\frac{8}{4 \sqrt{6}} is 63\frac{\sqrt{6}}{3}.

Q: How do I simplify a fraction with a square root in the denominator?

A: To simplify a fraction with a square root in the denominator, you need to follow a series of steps, including simplifying the fraction and the square root separately, combining them, and rationalizing the denominator.

Q: What is the conjugate of a square root?

A: The conjugate of a square root is the same square root itself.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to get rid of the square root in the denominator. This is because a square root is an irrational number, and we want to simplify the expression to its simplest form.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a square root is the same square root itself.

Q: What is the difference between simplifying an expression and rationalizing the denominator?

A: Simplifying an expression involves simplifying the fraction and the square root separately, combining them, and rationalizing the denominator. Rationalizing the denominator involves getting rid of the square root in the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q: Can I simplify an expression with a square root in the numerator?

A: Yes, you can simplify an expression with a square root in the numerator. However, you need to follow the same steps as simplifying an expression with a square root in the denominator.

Q: How do I simplify an expression with a square root in the numerator?

A: To simplify an expression with a square root in the numerator, you need to follow the same steps as simplifying an expression with a square root in the denominator. You need to simplify the fraction and the square root separately, combine them, and rationalize the denominator.

Q: What is the final simplified form of the expression 846\frac{8}{4 \sqrt{6}}?

A: The final simplified form of the expression 846\frac{8}{4 \sqrt{6}} is 63\frac{\sqrt{6}}{3}.

Q: Can I use a calculator to simplify an expression with a square root?

A: Yes, you can use a calculator to simplify an expression with a square root. However, it's always best to simplify the expression by hand to understand the steps involved.

Q: How do I check if my simplified expression is correct?

A: To check if your simplified expression is correct, you need to plug in the original expression and the simplified expression into a calculator and compare the results.

Real-World Applications

Simplifying expressions with square roots has many real-world applications. For example:

  • In physics, you need to simplify expressions with square roots to calculate the energy of a particle.
  • In engineering, you need to simplify expressions with square roots to calculate the stress on a material.
  • In finance, you need to simplify expressions with square roots to calculate the interest rate on a loan.

Conclusion

Simplifying expressions with square roots is an essential skill in mathematics. By following the steps outlined in this article, you can simplify expressions with square roots and get rid of the square root in the denominator. Remember to always rationalize the denominator to get the final simplified form of the expression.

Frequently Asked Questions

Q: What is the simplified form of the expression 846\frac{8}{4 \sqrt{6}}?

A: The simplified form of the expression 846\frac{8}{4 \sqrt{6}} is 63\frac{\sqrt{6}}{3}.

Q: How do I simplify a fraction with a square root in the denominator?

A: To simplify a fraction with a square root in the denominator, you need to follow a series of steps, including simplifying the fraction and the square root separately, combining them, and rationalizing the denominator.

Q: What is the conjugate of a square root?

A: The conjugate of a square root is the same square root itself.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to get rid of the square root in the denominator. This is because a square root is an irrational number, and we want to simplify the expression to its simplest form.

References