[tex]$27=x^{\frac{3}{2}}$[/tex]
Answer (2)
Raise both sides of the equation to the power of 3 2 : ( 27 ) 3 2 = ( x 2 3 ) 3 2 .
Simplify the exponents: x = 2 7 3 2 .
Rewrite 27 as 3 3 : x = ( 3 3 ) 3 2 .
Simplify and calculate: x = 3 3 ⋅ 3 2 = 3 2 = 9 . The final answer is 9 .
Explanation
Understanding the Problem We are given the equation 27 = x 2 3 and our goal is to find the value of x . This involves understanding fractional exponents and how to manipulate them. Let's break it down step by step!
Raising Both Sides to the Power of 2/3 To isolate x , we need to get rid of the exponent 2 3 . We can do this by raising both sides of the equation to the reciprocal of 2 3 , which is 3 2 . This will cancel out the exponent on x . So, we have: ( 27 ) 3 2 = ( x 2 3 ) 3 2
Simplifying the Exponents Now, let's simplify the exponents. When we raise a power to another power, we multiply the exponents. On the right side, we have: x 2 3 ⋅ 3 2 = x 1 = x So, our equation becomes: ( 27 ) 3 2 = x
Rewriting 27 as 3^3 We can rewrite 27 as 3 3 . This will help us simplify the left side of the equation: x = ( 3 3 ) 3 2
Simplifying the Expression Now, we simplify the expression on the left side by multiplying the exponents: x = 3 3 ⋅ 3 2 = 3 2
Calculating 3^2 Finally, we calculate 3 2 to find the value of x :
x = 3 2 = 9 So, the solution to the equation 27 = x 2 3 is x = 9 .
Examples
Imagine you are designing a garden and want to create a square-shaped area for planting flowers. If the area needs to be 27 square feet, and you know the side length is related to the area by the equation A = s 2 3 , where A is the area and s is the side length, solving this equation helps you determine the exact length of each side. Understanding and solving such equations allows you to accurately plan and construct various shapes and structures in real-world applications, ensuring precise dimensions and efficient use of space.
To solve 27 = x 2 3 , we raise both sides to the power of 3 2 to isolate x . Simplifying gives us x = 9 . Therefore, the final answer is x = 9 .
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