\(\left(\sqrt{x}^3\right)^{\frac{2}{3}}\)
Answer (1)
Rewrite the expression using fractional exponents: ( x 3 ) 3 2 = ( x 2 1 × 3 ) 3 2 = ( x 2 3 ) 3 2 .
Apply the power of a power rule: ( x a ) b = x a × b , so ( x 2 3 ) 3 2 = x 2 3 × 3 2 .
Simplify the exponent: x 2 3 × 3 2 = x 1 .
The simplified expression is: x .
Explanation
Understanding the Expression We are given the expression ( x 3 ) 3 2 and our goal is to simplify it. Let's break it down step by step. Remember that a square root can be represented as a fractional exponent, so x = x 2 1 .
Rewriting with Fractional Exponents First, let's rewrite the expression using fractional exponents: ( x 3 ) 3 2 = ( ( x 2 1 ) 3 ) 3 2 Now, we need to simplify the inner exponent. When raising a power to a power, we multiply the exponents: ( x 2 1 ) 3 = x 2 1 ⋅ 3 = x 2 3 So our expression becomes: ( x 2 3 ) 3 2
Applying the Power of a Power Rule Next, we apply the power of a power rule again. This rule states that ( x a ) b = x a ⋅ b . In our case, we have: ( x 2 3 ) 3 2 = x 2 3 ⋅ 3 2
Simplifying the Exponent Now, we simplify the exponent by multiplying the fractions: 2 3 ⋅ 3 2 = 2 ⋅ 3 3 ⋅ 2 = 6 6 = 1 So our expression simplifies to: x 1 = x
Final Answer Therefore, the simplified expression is just x .
x
Examples
Imagine you are calculating the area of a square, and you have a formula that involves nested exponents like the one we just simplified. Simplifying such expressions makes the calculations much easier and more straightforward. For example, if the side of a square is given by x 3 and you need to find the area by squaring it and then taking the 3 2 power, simplifying the expression first will give you the area directly as x .
