Raise to the power: [tex]$\left(3 x^2\right)^3$[/tex]
Answer (2)
To raise ( 3 x 2 ) to the power of 3, we use the power of a product rule and the power of a power rule, leading us to the final answer of 27 x 6 . By breaking down the components, we calculated 3 3 = 27 and ( x 2 ) 3 = x 6 . Combining these results gives 27 x 6 .
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Apply the power of a product rule: ( 3 x 2 ) 3 = 3 3 ( x 2 ) 3 .
Calculate 3 3 = 27 .
Apply the power of a power rule: ( x 2 ) 3 = x 6 .
Multiply the results: 27 x 6 , so the final answer is 27 x 6 .
Explanation
Understanding the Problem We are asked to raise the expression ( 3 x 2 ) to the power of 3. This means we need to compute ( 3 x 2 ) 3 . To do this, we'll use the power of a product rule and the power of a power rule.
Applying the Power of a Product Rule First, we apply the power of a product rule, which states that ( ab ) n = a n b n . In our case, a = 3 and b = x 2 , and n = 3 . So we have: ( 3 x 2 ) 3 = 3 3 ⋅ ( x 2 ) 3
Applying the Power of a Power Rule Next, we apply the power of a power rule, which states that ( a m ) n = a mn . In our case, we have ( x 2 ) 3 = x 2 ⋅ 3 = x 6 . Also, we calculate 3 3 = 3 ⋅ 3 ⋅ 3 = 27 .
Multiplying the Results Now, we multiply the results together: 3 3 ⋅ ( x 2 ) 3 = 27 ⋅ x 6 = 27 x 6
Examples
Understanding how to raise expressions to a power is useful in many areas, such as calculating the volume of a cube. If the side of a cube is 3 x 2 , then the volume of the cube is ( 3 x 2 ) 3 = 27 x 6 . This concept is also fundamental in physics, engineering, and computer graphics, where scaling and exponential growth are frequently encountered.
