Solve the Product: Seventh Root of 4 Times Cube Root of 4

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

Converting radicals to fractional exponents lets you use the multiplication rule for same bases! $ \sqrt[n]{a} = a^{\frac{1}{n}} $ makes complex radical operations much simpler.

Q: How do I add fractions like 1/7 + 1/3?

Find the common denominator: $ \frac{1}{7} + \frac{1}{3} = \frac{3}{21} + \frac{7}{21} = \frac{10}{21} $. So the answer is $ 4^{\frac{10}{21}} $!

Q: Can I simplify the radical further?

The expression $ 4^{\frac{1}{7}+\frac{1}{3}} $ is the simplified form using exponent laws. You could write it as $ \sqrt[21]{4^{10}} $, but that's more complicated!

Q: What if the bases were different numbers?

If you had $ \sqrt[7]{4} \cdot \sqrt[3]{8} $, you cannot combine them directly. The multiplication rule $ a^m \cdot a^n = a^{m+n} $ only works with identical bases.

Q: Why isn't the answer just 4^(1/7)?

That would ignore the $ \sqrt[3]{4} $ part completely! When multiplying, you must add the exponents: $ \frac{1}{7} + \frac{1}{3} \neq \frac{1}{7} $.

Radical Multiplication with Exponent Laws

Solve the following exercise:

$\sqrt[7]{4}\cdot\sqrt[3]{4}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following expression

00:03 The C root of the A value to the power of B

00:06 The result will be equal to number A to the power of B divided by C

00:09 Every number is essentially to the power of 1

00:12 We will use this formula in our exercise

00:16 When multiplying powers with equal bases

00:19 The power of the result equals the sum of the powers

00:22 We will use this formula in our exercise, and add the powers

00:25 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt[7]{4}\cdot\sqrt[3]{4}=$

Step-by-step solution

In order to simplify the given expression, we will use two laws of exponents:

a. The definition of root as an exponent:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

b. The law of exponents for multiplication between terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Let's start by converting the roots to exponents using the law mentioned in a':

$\sqrt[\textcolor{red}{7}]{4}\cdot\sqrt[\textcolor{blue}{3}]{4}= \\ \downarrow\\ 4^{\frac{1}{\textcolor{red}{7}}}\cdot4^{\frac{1}{\textcolor{blue}{3}}}=$

We'll continue, since we are multiplying two terms with identical bases - we'll use the law of exponents mentioned in b':

$4^{\frac{1}{7}}\cdot4^{\frac{1}{3}}= \\ \boxed{4^{\frac{1}{7}+\frac{1}{3}}}$

Therefore, the correct answer is answer b'.

Final Answer

$4^{\frac{1}{7}+\frac{1}{3}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert radicals to fractional exponents before multiplying
Technique: $\sqrt[7]{4} = 4^{\frac{1}{7}}$ and $\sqrt[3]{4} = 4^{\frac{1}{3}}$
Check: Same base multiplication adds exponents: $\frac{1}{7} + \frac{1}{3} = \frac{10}{21}$ ✓

Common Mistakes

Avoid these frequent errors

Adding the root indices instead of the fractional exponents
Don't add 7 + 3 = 10 to get $4^{10}$ = huge wrong answer! The root indices become denominators in fractional exponents, not the exponents themselves. Always convert to $4^{\frac{1}{7}} \cdot 4^{\frac{1}{3}}$ then add $\frac{1}{7} + \frac{1}{3}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to convert radicals to exponents?

Converting radicals to fractional exponents lets you use the multiplication rule for same bases! $\sqrt[n]{a} = a^{\frac{1}{n}}$ makes complex radical operations much simpler.

How do I add fractions like 1/7 + 1/3?

Find the common denominator: $\frac{1}{7} + \frac{1}{3} = \frac{3}{21} + \frac{7}{21} = \frac{10}{21}$ . So the answer is $4^{\frac{10}{21}}$ !

Can I simplify the radical further?

The expression $4^{\frac{1}{7}+\frac{1}{3}}$ is the simplified form using exponent laws. You could write it as $\sqrt[21]{4^{10}}$ , but that's more complicated!

What if the bases were different numbers?

If you had $\sqrt[7]{4} \cdot \sqrt[3]{8}$ , you cannot combine them directly. The multiplication rule $a^m \cdot a^n = a^{m+n}$ only works with identical bases.

Why isn't the answer just 4^(1/7)?

That would ignore the $\sqrt[3]{4}$ part completely! When multiplying, you must add the exponents: $\frac{1}{7} + \frac{1}{3} \neq \frac{1}{7}$ .

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Solve the Product: Seventh Root of 4 Times Cube Root of 4

Radical Multiplication with Exponent Laws

❤️ 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 exponents?

How do I add fractions like 1/7 + 1/3?

Can I simplify the radical further?

What if the bases were different numbers?

Why isn't the answer just 4^(1/7)?

🌟 Unlock Your Math Potential

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