%5E3%2F2-in-radical-form.jpeg "(10n)^3/2 In Radical Form")
Simplifying $(10n)^{3/2}$ in Radical Form
The expression $(10n)^{3/2}$ represents a power with a fractional exponent. Understanding how to work with these types of exponents is essential in simplifying radical expressions.
Understanding Fractional Exponents
A fractional exponent like 3/2 can be broken down into two parts:
- The numerator (3): This indicates the power to which the base is raised.
- The denominator (2): This indicates the root to be taken.
Therefore, $(10n)^{3/2}$ is equivalent to taking the square root of $(10n)$ cubed.
Simplifying the Expression
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Cube the base: $(10n)^3 = 10^3 * n^3 = 1000n^3$
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Take the square root: $\sqrt{1000n^3}$
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Simplify the radical: We can factor out the perfect square from the radicand.
- $\sqrt{1000n^3} = \sqrt{100 * 10 * n^2 * n} = \sqrt{100n^2} * \sqrt{10n} = \boxed{10n\sqrt{10n}}$
Therefore, the simplified radical form of $(10n)^{3/2}$ is 10n√(10n).