Simplify. [tex]\sqrt{6 z^6} \sqrt{2 z}[/tex]
Assume that the variable represents a positive real number.
Answer (2)
Combine the square roots: 6 z 6 2 z = ( 6 z 6 ) ( 2 z ) .
Simplify the expression inside the square root: ( 6 z 6 ) ( 2 z ) = 12 z 7 .
Factor out perfect squares: 12 z 7 = ( 4 z 6 ) ( 3 z ) .
Simplify the expression: 12 z 7 = 2 z 3 3 z .
2 z 3 3 z
Explanation
Understanding the Problem We are given the expression 6 z 6 2 z and we want to simplify it. We assume that z is a positive real number.
Combining Square Roots We can use the property a b = ab to combine the two square roots: 6 z 6 2 z = ( 6 z 6 ) ( 2 z )
Simplifying the Expression Now, we simplify the expression inside the square root: ( 6 z 6 ) ( 2 z ) = 12 z 7 So we have 12 z 7 .
Factoring Perfect Squares We can factor out perfect squares from the expression inside the square root: 12 z 7 = 4 × 3 × z 6 × z = ( 4 z 6 ) × ( 3 z )
Rewriting the Square Root Now we rewrite the square root as 12 z 7 = 4 z 6 3 z
Simplifying Perfect Square We simplify the square root of the perfect square: 4 z 6 = 2 z 3
Final Simplified Expression Finally, we write the simplified expression: 2 z 3 3 z
Examples
Imagine you are calculating the area of a rectangular garden where one side's length is expressed as 6 z 6 and the other side as 2 z . Simplifying the expression 6 z 6 2 z allows you to find a more manageable form, 2 z 3 3 z , to easily compute the garden's area for different values of z . This simplification helps in practical applications where complex expressions need to be evaluated quickly and efficiently, such as in engineering or physics calculations.
The simplified form of 6 z 6 2 z is 2 z 3 3 z . This involves combining the square roots and factoring out perfect squares. Each step follows basic properties of square roots and algebraic manipulation.
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