Simplify √25 · ∛(√25): Multiple Radical Expression Problem

Q: Why do I need to convert radicals to fractional exponents?

Fractional exponents make it much easier to apply exponent laws! $ \sqrt{25} = 25^{1/2} $ and $ \sqrt[3]{\sqrt{25}} = 25^{1/6} $ can be multiplied using addition of exponents.

Q: How do I handle the nested radical $ \sqrt[3]{\sqrt{25}} $ ?

Work from the inside out: First convert $ \sqrt{25} = 25^{1/2} $, then apply the cube root: $ \sqrt[3]{25^{1/2}} = (25^{1/2})^{1/3} = 25^{1/6} $.

Q: Why does $ 25^{2/3} = 5^{4/3} $ ?

Since $ 25 = 5^2 $, we can substitute: $ 25^{2/3} = (5^2)^{2/3} = 5^{2 \cdot 2/3} = 5^{4/3} $. This uses the power rule $ (a^m)^n = a^{mn} $.

Q: How do I convert $ 5^{4/3} $ to mixed number form?

Divide the exponent: $ 4 ÷ 3 = 1 $ remainder $ 1 $, so $ 5^{4/3} = 5^{1 + 1/3} = 5^{1\frac{1}{3}} $. The mixed number form is often preferred in final answers.

Q: Can I solve this without using exponent laws?

It's much more difficult! You'd need to calculate $ \sqrt[3]{\sqrt{25}} = \sqrt[3]{5} $ and then find $ 5 \times \sqrt[3]{5} $. Using exponent laws gives a cleaner, more systematic approach.

Radical Expressions with Nested Roots

Complete the following exercise:

$\sqrt{25}\cdot\sqrt[3]{\sqrt{25}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:08 A "regular" root is of the order 2

00:11 When there is a root of the order (C) for root (B)

00:15 The result equals the root of the product of the orders

00:18 We will apply this formula to our exercise

00:30 We'll break down 25 to 5 squared

00:38 When we have a root of the order (C) on a number (A) to the power of (B)

00:41 The result equals number (A) to the power of (B divided by C)

00:45 We will apply this formula to our exercise

00:51 Calculate the exponent portions

01:01 When we have a multiplication between powers with equal bases

01:04 The result equals the base with an exponent equal to the sum of the powers

01:07 We will apply this formula to our exercise

01:12 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt{25}\cdot\sqrt[3]{\sqrt{25}}=$

Step-by-step solution

To solve the problem $\sqrt{25} \cdot \sqrt[3]{\sqrt{25}}$ , follow these steps:

First, express each root in terms of exponents:

$\sqrt{25} = 25^{1/2}$
$\sqrt[3]{\sqrt{25}} = \sqrt[3]{25^{1/2}}$

Using the law of exponents $(a^m)^n = a^{m \cdot n}$ , simplify $\sqrt[3]{25^{1/2}}$ as follows:

$(25^{1/2})^{1/3} = 25^{(1/2) \cdot (1/3)} = 25^{1/6}$

Now, multiply the two expressions:

$25^{1/2} \cdot 25^{1/6} = 25^{(1/2 + 1/6)}$
Calculate the sum of the exponents: $\frac{1}{2} + \frac{1}{6} = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$
This gives: $25^{2/3}$

Recognize $25 = 5^2$ , thus: $(5^2)^{2/3} = 5^{2 \cdot (2/3)} = 5^{4/3}$
Convert mixed fraction: $5^{4/3} = 5^{1 + 1/3} = 5^{1\frac{1}{3}}$

Therefore, the product $\sqrt{25} \cdot \sqrt[3]{\sqrt{25}}$ equals $\mathbf{5^{1\frac{1}{3}}}$ .

Final Answer

$5^{1\frac{1}{3}}$

Key Points to Remember

Essential concepts to master this topic

Conversion: Express all radicals as fractional exponents first
Technique: Apply $(a^m)^n = a^{m \cdot n}$ to simplify $\sqrt[3]{\sqrt{25}}$
Check: Verify $5^{4/3} = 5^{1\frac{1}{3}}$ by calculating $5^{4/3} = \sqrt[3]{5^4} = \sqrt[3]{625}$ ✓

Common Mistakes

Avoid these frequent errors

Calculating nested radicals from the outside in
Don't evaluate $\sqrt{25} = 5$ first then try to multiply by $\sqrt[3]{5}$ = wrong approach! This makes the problem much harder and leads to incorrect simplification. Always convert all radicals to fractional exponents first, then use exponent laws to combine them systematically.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{30}\cdot\sqrt{1}=$

FAQ

Everything you need to know about this question

Why do I need to convert radicals to fractional exponents?

Fractional exponents make it much easier to apply exponent laws! $\sqrt{25} = 25^{1/2}$ and $\sqrt[3]{\sqrt{25}} = 25^{1/6}$ can be multiplied using addition of exponents.

How do I handle the nested radical $\sqrt[3]{\sqrt{25}}$ ?

Work from the inside out: First convert $\sqrt{25} = 25^{1/2}$ , then apply the cube root: $\sqrt[3]{25^{1/2}} = (25^{1/2})^{1/3} = 25^{1/6}$ .

Why does $25^{2/3} = 5^{4/3}$ ?

Since $25 = 5^2$ , we can substitute: $25^{2/3} = (5^2)^{2/3} = 5^{2 \cdot 2/3} = 5^{4/3}$ . This uses the power rule $(a^m)^n = a^{mn}$ .

How do I convert $5^{4/3}$ to mixed number form?

Divide the exponent: $4 ÷ 3 = 1$ remainder $1$ , so $5^{4/3} = 5^{1 + 1/3} = 5^{1\frac{1}{3}}$ . The mixed number form is often preferred in final answers.

Can I solve this without using exponent laws?

It's much more difficult! You'd need to calculate $\sqrt[3]{\sqrt{25}} = \sqrt[3]{5}$ and then find $5 \times \sqrt[3]{5}$ . Using exponent laws gives a cleaner, more systematic approach.

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Simplify √25 · ∛(√25): Multiple Radical Expression Problem

Radical Expressions with Nested Roots

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to convert radicals to fractional exponents?

How do I handle the nested radical $\sqrt[3]{\sqrt{25}}$ ?

Why does $25^{2/3} = 5^{4/3}$ ?

How do I convert $5^{4/3}$ to mixed number form?

Can I solve this without using exponent laws?

🌟 Unlock Your Math Potential

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Simplify √25 · ∛(√25): Multiple Radical Expression Problem

Radical Expressions with Nested Roots

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to convert radicals to fractional exponents?

How do I handle the nested radical 253 \sqrt[3]{\sqrt{25}} 325​​?

Why does 252/3=54/3 25^{2/3} = 5^{4/3} 252/3=54/3?

How do I convert 54/3 5^{4/3} 54/3 to mixed number form?

Can I solve this without using exponent laws?

🌟 Unlock Your Math Potential

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Solve Nested Square Roots: √√49 × √√16 Multiplication Problem

Simplify the Expression: Cube Root of √25 × Cube Root of √64

Simplify the Expression: Cube Root of Square Root of 64 × Cube Root of 64

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How do I handle the nested radical $\sqrt[3]{\sqrt{25}}$ ?

Why does $25^{2/3} = 5^{4/3}$ ?

How do I convert $5^{4/3}$ to mixed number form?