Answer:
10)
[tex]=-4\sqrt[3]{2}[/tex]
12)
[tex]=3\sqrt[3]6[/tex]
14)
[tex]=-10x\sqrt[3]{x}[/tex]
Step-by-step explanation:
10)
We have:
[tex]\sqrt[3]{-128}[/tex]
We want to simplify the cube root. So, let's start picking at the factors.
Notice that -128 is divisible by 64. So:
[tex]=\sqrt[3]{-1\cdot64\cdot2}[/tex]
Now, notice that 64 is the same as 4³. (-1) is also the same as (-1)³. Therefore:
[tex]=\sqrt{(-1)^3\cdot(4)^3\cdot2[/tex]
We can now expand our root:
[tex]=\sqrt[3]{(-1)^3}\cdot\sqrt[3]{(4)^3}\cdot\sqrt[3]{2}[/tex]
The cubes and the cube roots cancel each other out. This leaves us with:
[tex]=(-1)\cdot(4)\cdot\sqrt[3]2[/tex]
Simplify:
[tex]=-4\sqrt[3]{2}[/tex]
12)
We have:
[tex]\sqrt[3]{162}[/tex]
Again, let's see what we can factor out.
If it's unclear what we can factor out, we can guess and check. Since 162 is even, let's divide it by 2. This yields:
[tex]=\sqrt[3]{2\cdot81}[/tex]
Notice here that 81 is the same as 3⁴. Therefore:
[tex]=\sqrt[3]{2\cdot3^4}[/tex]
We can separate the 3 from the exponent using the properties of exponents:
[tex]=\sqrt[3]{2\cdot3\cdot3^3}[/tex]
Expand:
[tex]=\sqrt[3]{3^3}\cdot\sqrt[3]{2\cdot3}[/tex]
The cube roots and cube will cancel. This leaves us with:
[tex]=3\cdot\sqrt{2\cdot3}[/tex]
Simplify:
[tex]=3\sqrt[3]6[/tex]
14)
We have:
[tex]\sqrt[3]{-1000x^4}[/tex]
Notice here that 1000 is the same as 10³. Also, we can separate an x from the x⁴. Therefore:
[tex]=\sqrt[3]{-1\cdot (10)^3\cdot x^3\cdot x}[/tex]
Also, (-1) is the same as (-1)³. Thus:
[tex]=\sqrt[3]{(-1)^3\cdot (10)^3\cdot x^3\cdot x}[/tex]
Expand:
[tex]=\sqrt[3]{(-1)^3}\cdot\sqrt[3]{(10)^3}\cdot\sqrt[3]{x^3}\cdot\sqrt[3]{x}[/tex]
The cube roots and the cube will cancel. This leaves us with:
[tex]=-1\cdot10\cdot x\cdot\sqrt[3]{x}[/tex]
Simplify:
[tex]=-10x\sqrt[3]{x}[/tex]
And we're done!
