What is the simplified form of the following expression?
[tex]$\sqrt[3]{\frac{4 x}{5}}$[/tex]
A. [tex]$\frac{\sqrt[3]{4 x}}{5}$[/tex]
B. [tex]$\frac{\sqrt[3]{20 x}}{5}$[/tex]
C. [tex]$\frac{\sqrt[3]{100 x}}{5}$[/tex]
D. [tex]$\frac{\sqrt{100 x}}{125}$[/tex]
Answer (2)
The simplified form of the expression 3 5 4 x is 5 3 100 x , which matches option C in the choices provided.
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Rewrite the expression as a fraction of cube roots: 3 5 4 x = 3 5 3 4 x .
Rationalize the denominator by multiplying both numerator and denominator by 3 25 .
Simplify the expression to 3 125 3 100 x .
Since 3 125 = 5 , the simplified form is 5 3 100 x .
Explanation
Understanding the Problem We are given the expression 3 5 4 x . Our goal is to simplify this expression and see which of the given options matches our simplified form.
Rationalizing the Denominator We can rewrite the given expression as a fraction of cube roots: 3 5 4 x = 3 5 3 4 x .To rationalize the denominator, we want to get rid of the cube root in the denominator. We can do this by multiplying both the numerator and the denominator by a factor that will make the denominator a perfect cube. Since we have 3 5 in the denominator, we need to multiply by 3 5 2 = 3 25 to get 3 5 3 = 3 125 = 5 .
Multiplying by the Rationalizing Factor Multiply both the numerator and the denominator by 3 25 : 3 5 3 4 x ⋅ 3 25 3 25 = 3 5 ⋅ 3 25 3 4 x ⋅ 3 25 = 3 5 ⋅ 25 3 4 x ⋅ 25 = 3 125 3 100 x .
Simplifying the Cube Root Since 3 125 = 5 , we can simplify the expression further: 3 125 3 100 x = 5 3 100 x .
Matching with the Given Options Now, we compare our simplified expression 5 3 100 x with the given options: 5 3 4 x , 5 3 20 x , 5 3 100 x , 125 100 x .We see that our simplified expression matches the third option: 5 3 100 x .
Examples
Cube roots are used in various fields, such as engineering and physics, to calculate volumes and dimensions. For example, if you have a cube-shaped container and you know its volume, you can use the cube root to find the length of one side. This is also applicable in architecture when designing structures with specific volume requirements. Understanding how to simplify expressions with cube roots helps in making these calculations more manageable and accurate.
