Sure! Let's simplify the expression [tex]\(\sqrt[3]{50 a ^2 b^4} \times \sqrt[3]{20 a ^7 b^2}\)[/tex] step-by-step.
### Step 1: Combine the Radicals
First, we can use the property of radicals that states the product of cube roots is the cube root of the product. This means we can combine the two cube roots into one:
[tex]\[
\sqrt[3]{50 a^2 b^4} \times \sqrt[3]{20 a^7 b^2} = \sqrt[3]{(50 a^2 b^4) \times (20 a^7 b^2)}
\][/tex]
### Step 2: Multiply Inside the Radicals
Now, we multiply the terms inside the radicals:
[tex]\[
(50 a^2 b^4) \times (20 a^7 b^2) = 50 \times 20 \times a^2 \times a^7 \times b^4 \times b^2
\][/tex]
### Step 3: Multiply the Constants
First, we multiply the constant numbers:
[tex]\[
50 \times 20 = 1000
\][/tex]
### Step 4: Combine Like Terms
Next, we use the properties of exponents to combine the like variables. For the [tex]\(a\)[/tex] terms and [tex]\(b\)[/tex] terms, we add the exponents together:
[tex]\[
a^2 \times a^7 = a^{2+7} = a^9
\][/tex]
[tex]\[
b^4 \times b^2 = b^{4+2} = b^6
\][/tex]
### Step 5: Form the New Radical
Now we have:
[tex]\[
1000 \times a^9 \times b^6
\][/tex]
So, we can write our combined radical as:
[tex]\[
\sqrt[3]{1000 a^9 b^6}
\][/tex]
### Step 6: Simplify the Cube Root
Finally, we simplify the cube root of each of the factors inside:
- The cube root of 1000:
[tex]\[
\sqrt[3]{1000} = 10 \quad \text{(since \(10^3 = 1000\))}
\][/tex]
- The cube root of [tex]\(a^9\)[/tex]:
[tex]\[
\sqrt[3]{a^9} = a^{9/3} = a^3
\][/tex]
- The cube root of [tex]\(b^6\)[/tex]:
[tex]\[
\sqrt[3]{b^6} = b^{6/3} = b^2
\][/tex]
### Step 7: Combine the Simplified Terms
Putting it all together, we have:
[tex]\[
\sqrt[3]{1000 a^9 b^6} = 10 \times a^3 \times b^2
\][/tex]
So the simplified form of the original expression is:
[tex]\[
10 a^3 b^2
\][/tex]
Thus, [tex]\(\sqrt[3]{50 a ^2 b^4} \times \sqrt[3]{20 a ^7 b^2} = 10 a^3 b^2\)[/tex].
