Simplifying the Square Root Term

Oct 13, 2025 by miodeya.com 86 views

**Which Expression is Equivalent to $$\frac{\sqrt{10}}{\sqrt[4]{8}}$$?**

Understanding the Problem

The given expression is $$\frac{\sqrt{10}}{\sqrt[4]{8}}$$. To simplify this expression, we need to find an equivalent expression that is easier to evaluate. This involves simplifying the square root and fourth root terms.

Simplifying the Square Root Term

The square root term in the expression is $$\sqrt{10}$$. We can simplify this term by expressing it as a product of two square roots. Since $$\sqrt{10} = \sqrt{2 \times 5}$$, we can rewrite it as $$\sqrt{2} \times \sqrt{5}$$.

Simplifying the Fourth Root Term

The fourth root term in the expression is $$\sqrt[4]{8}$$. We can simplify this term by expressing it as a product of two square roots. Since $$\sqrt[4]{8} = \sqrt{\sqrt{8}}$$, we can rewrite it as $$\sqrt{2}$$.

Substituting the Simplified Terms

Now that we have simplified the square root and fourth root terms, we can substitute them back into the original expression. This gives us $$\frac{\sqrt{2} \times \sqrt{5}}{\sqrt{2}}$$.

Cancelling Out the Common Term

We can simplify the expression further by cancelling out the common term $$\sqrt{2}$$. This leaves us with $$\frac{\sqrt{5}}{1}$$, which simplifies to $$\sqrt{5}$$.

Comparing the Simplified Expression with the Options

Now that we have simplified the expression, we can compare it with the options provided. The simplified expression is $$\sqrt{5}$$, which is equivalent to option C. $$\frac{2 \sqrt{5}}{5}$$.

Conclusion

In conclusion, the expression $$\frac{\sqrt{10}}{\sqrt[4]{8}}$$ is equivalent to option C. $$\frac{2 \sqrt{5}}{5}$$.

Frequently Asked Questions

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

A: A square root is a number that, when multiplied by itself, gives the original number. For example, $$\sqrt{16} = 4$$ because $$4 \times 4 = 16$$. A fourth root is a number that, when raised to the power of 4, gives the original number. For example, $$\sqrt[4]{16} = 2$$ because $$2^4 = 16$$.

Q: How do you simplify a square root term?

A: To simplify a square root term, you can express it as a product of two square roots. For example, $$\sqrt{10} = \sqrt{2 \times 5} = \sqrt{2} \times \sqrt{5}$$.

Q: How do you simplify a fourth root term?

A: To simplify a fourth root term, you can express it as a product of two square roots. For example, $$\sqrt[4]{8} = \sqrt{\sqrt{8}} = \sqrt{2}$$.

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

A: A rational expression is an expression that can be simplified to a rational number, which is a number that can be expressed as a ratio of two integers. For example, $$\frac{1}{2}$$ is a rational number. An irrational expression is an expression that cannot be simplified to a rational number. For example, $$\sqrt{2}$$ is an irrational number.

Q: How do you compare two expressions?

A: To compare two expressions, you can simplify them and then compare the simplified expressions. For example, if we have two expressions $$\frac{\sqrt{10}}{\sqrt[4]{8}}$$ and $$\frac{2 \sqrt{5}}{5}$$, we can simplify the first expression to $$\sqrt{5}$$ and then compare it with the second expression.

Q: What is the final answer to the problem?

A: The final answer to the problem is option C. $$\frac{2 \sqrt{5}}{5}$$.

Q: Why is the final answer option C?

A: The final answer is option C because the simplified expression $$\sqrt{5}$$ is equivalent to option C. $$\frac{2 \sqrt{5}}{5}$$.

Q: Can you provide more examples of simplifying expressions?

A: Yes, here are a few more examples of simplifying expressions:

$\frac{\sqrt{20}}{\sqrt{5}} = \frac{\sqrt{4 \times 5}}{\sqrt{5}} = \frac{2 \sqrt{5}}{\sqrt{5}} = 2$
$\frac{\sqrt[4]{16}}{\sqrt{2}} = \frac{\sqrt[4]{2^4}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \frac{2 \sqrt{2}}{2} = \sqrt{2}$

These examples demonstrate how to simplify expressions by cancelling out common terms and using the properties of square roots and fourth roots.