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*Algebra 2 Practice - Multiplying and Simplifying Cubed Roots*

To multiply and simplify cubed roots, follow these steps:

---

*Step 1: Multiply the Radicands*
When multiplying cubed roots, multiply the radicands (the numbers inside the cube roots) first. The general rule is:

\[
\sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b}
\]

---

*Step 2: Simplify the Cube Root*
After multiplying the radicands, simplify the resulting cube root by factoring the radicand into its prime factors. Look for perfect cubes (like \(8 = 2^3\), \(27 = 3^3\), etc.), and simplify the cube root by extracting the perfect cubes.

---

*Example 1: Multiply and Simplify*

Multiply the following cubed roots and simplify:

\[
\sqrt[3]{4} \times \sqrt[3]{16}
\]

1. **Multiply the Radicands**:
Multiply the numbers inside the cube roots:
\[
4 \times 16 = 64
\]
So, we now have:
\[
\sqrt[3]{64}
\]

2. **Simplify the Cube Root**:
The cube root of 64 is \(4\), because \(4^3 = 64\). So, the simplified result is:
\[
\sqrt[3]{64} = 4
\]

---

*Example 2: Multiply and Simplify with Variables*

Multiply and simplify the following cube roots:

\[
\sqrt[3]{8x^3} \times \sqrt[3]{27y^6}
\]

1. **Multiply the Radicands**:
Multiply the numbers and variables inside the cube roots:
\[
(8x^3) \times (27y^6) = 8 \times 27 \times x^3 \times y^6 = 216x^3y^6
\]

2. **Simplify the Cube Root**:
Now simplify \( \sqrt[3]{216x^3y^6} \):
\( 216 = 6^3 \), so \( \sqrt[3]{216} = 6 \)
\( x^3 \) is a perfect cube, so \( \sqrt[3]{x^3} = x \)
\( y^6 = (y^3)^2 \), so \( \sqrt[3]{y^6} = y^2 \)

Putting it all together:
\[
\sqrt[3]{216x^3y^6} = 6xy^2
\]

---

*Step 3: Final Answer*

The simplified result for \( \sqrt[3]{8x^3} \times \sqrt[3]{27y^6} \) is:

\[
6xy^2
\]

---

**General Rule for Simplifying Cube Roots**:
Multiply the radicands (numbers inside the cube roots).
Look for perfect cubes and extract them.
Simplify the resulting expression by pulling out cube roots of perfect cubes.

---

**Summary**:
When multiplying cubed roots, multiply the radicands and simplify by extracting perfect cubes.
Apply this process to both numbers and variables inside the cube root.

I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out:

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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