Write the expression as a square of a monomial:
Example: [tex]$81 x^4=\left(9 x^2\right)^2$[/tex]
[tex]$0.09 y^{12}$[/tex]
Answer (2)
Find the square root of the coefficient: 0.09 = 0.3 .
Find the square root of the variable part: y 12 = y 6 .
Write the expression as a square of a monomial: ( 0.3 y 6 ) 2 .
The expression 0.09 y 12 as a square of a monomial is ( 0.3 y 6 ) 2 .
Explanation
Understanding the Problem We are given the expression 0.09 y 12 and asked to write it as a square of a monomial. This means we want to find an expression of the form ( A y B ) 2 that is equal to 0.09 y 12 .
Finding the Square Root of the Coefficient To do this, we need to find the square root of both the coefficient and the variable part of the expression. The square root of 0.09 is 0.3, since 0.3 × 0.3 = 0.09 .
Finding the Square Root of the Variable Part The square root of y 12 is y 6 , since ( y 6 ) 2 = y 6 × 2 = y 12 .
Writing the Expression as a Square of a Monomial Therefore, we can write 0.09 y 12 as ( 0.3 y 6 ) 2 .
Final Answer Thus, the expression 0.09 y 12 can be written as the square of the monomial 0.3 y 6 .
Examples
In physics, kinetic energy is often expressed as K E = 2 1 m v 2 , where m is the mass and v is the velocity. If you know the kinetic energy and mass, finding the velocity involves taking the square root. Rewriting expressions as squares helps simplify such calculations. For instance, if you have an expression like 49 x 2 representing some physical quantity, recognizing it as ( 7 x ) 2 can simplify further analysis or calculations, such as determining the velocity in a kinetic energy problem or simplifying equations in mechanics.
The expression 0.09 y 12 can be rewritten as the square of a monomial by taking the square root of the coefficient and the variable part. Thus, it can be expressed as ( 0.3 y 6 ) 2 . This shows the coefficients and the variable are neatly combined into one squared term.
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