Solve Nested Roots: Simplifying ⁷√(⁴√14) Step by Step

Q: Why do we multiply the indices instead of adding them?

Think of it like exponent rules in reverse! When you have $ \sqrt[7]{\sqrt[4]{14}} $, you're raising 14 to the power $ \frac{1}{4} $, then to the power $ \frac{1}{7} $. Multiplying exponents gives $ \frac{1}{4} \times \frac{1}{7} = \frac{1}{28} $.

Q: Can I simplify this further since we have 14 inside?

Not really! Since 14 = 2 × 7, we could write $ \sqrt[28]{2 \times 7} $, but this doesn't simplify nicely. The answer $ \sqrt[28]{14} $ is already in its simplest radical form.

Q: What if the inner root was something like √16 instead?

Great question! If you had $ \sqrt[7]{\sqrt{16}} $, you could first simplify $ \sqrt{16} = 4 $, then get $ \sqrt[7]{4} $. Always simplify inner expressions first when possible!

Q: How do I remember which operation to use with the indices?

Remember: "Nested roots multiply, nested powers add!" For roots within roots, multiply the indices. For powers of powers like $ (x^a)^b $, you add the exponents.

Q: Is there a way to check my answer without a calculator?

Yes! Work backwards using the definition of roots. If your answer is correct, then $ (\sqrt[28]{14})^{28} $ should equal 14. You can also verify the rule with simpler numbers first.

Nested Radicals with Root Multiplication Rule

Solve the following exercise:

$\sqrt[7]{\sqrt[4]{14}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:05 Let's solve this problem together.

00:09 Consider a number, A, raised to the power of B, inside a root of order C.

00:14 The result is A to the power of B divided by C.

00:19 Now, let's use this formula on our example.

00:23 First, find the multiplication of the order.

00:26 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt[7]{\sqrt[4]{14}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the inner and outer roots.
Step 2: Apply the root of a root formula.
Step 3: Calculate the combined root.

Now, let's work through each step:

Step 1: The expression given is $\sqrt[7]{\sqrt[4]{14}}$ . The inner root is $\sqrt[4]{14}$ , and the outer root is $\sqrt[7]{(\cdot)}$ .

Step 2: Use the formula for the root of a root: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$ .

Step 3: Plug in our values: $n = 7$ and $m = 4$ . Thus, we have:

$\sqrt[7]{\sqrt[4]{14}} = \sqrt[7 \times 4]{14} = \sqrt[28]{14}$

Therefore, the simplified form of the given expression is $\sqrt[28]{14}$ .

Final Answer

$\sqrt[28]{14}$

Key Points to Remember

Essential concepts to master this topic

Rule: When nesting roots, multiply the indices: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \times m]{x}$
Technique: Calculate indices first: 7 × 4 = 28, then $\sqrt[28]{14}$
Check: Verify by working backwards: $(\sqrt[28]{14})^{28} = 14$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of multiplying the root indices
Don't calculate 7 + 4 = 11 to get $\sqrt[11]{14}$ ! This ignores how nested operations work in mathematics. Always multiply the indices: 7 × 4 = 28 for the correct answer $\sqrt[28]{14}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

FAQ

Everything you need to know about this question

Why do we multiply the indices instead of adding them?

Think of it like exponent rules in reverse! When you have $\sqrt[7]{\sqrt[4]{14}}$ , you're raising 14 to the power $\frac{1}{4}$ , then to the power $\frac{1}{7}$ . Multiplying exponents gives $\frac{1}{4} \times \frac{1}{7} = \frac{1}{28}$ .

Can I simplify this further since we have 14 inside?

Not really! Since 14 = 2 × 7, we could write $\sqrt[28]{2 \times 7}$ , but this doesn't simplify nicely. The answer $\sqrt[28]{14}$ is already in its simplest radical form.

What if the inner root was something like √16 instead?

Great question! If you had $\sqrt[7]{\sqrt{16}}$ , you could first simplify $\sqrt{16} = 4$ , then get $\sqrt[7]{4}$ . Always simplify inner expressions first when possible!

How do I remember which operation to use with the indices?

Remember: "Nested roots multiply, nested powers add!" For roots within roots, multiply the indices. For powers of powers like $(x^a)^b$ , you add the exponents.

Is there a way to check my answer without a calculator?

Yes! Work backwards using the definition of roots. If your answer is correct, then $(\sqrt[28]{14})^{28}$ should equal 14. You can also verify the rule with simpler numbers first.

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