Math  /  Numbers & Operations

Question4) 512m3\sqrt{512 m^{3}}

Studdy Solution

STEP 1

1. We are given the expression 512m3\sqrt{512 m^3}.
2. We need to simplify this expression.

STEP 2

1. Simplify the numerical part under the square root.
2. Simplify the variable part under the square root.
3. Combine the simplified parts.

STEP 3

First, simplify the numerical part, 512 \sqrt{512} .
- Factor 512 512 into its prime factors: 512=29 512 = 2^9 .

STEP 4

Simplify the variable part, m3 \sqrt{m^3} .
- Use the property of square roots: m3=m2m=mm\sqrt{m^3} = \sqrt{m^2 \cdot m} = m \cdot \sqrt{m}.

STEP 5

Combine the simplified numerical and variable parts.
- Simplify 29 \sqrt{2^9} : $ \sqrt{2^9} = \sqrt{(2^4)^2 \cdot 2} = 2^4 \cdot \sqrt{2} = 16\sqrt{2} \]
- Combine with the variable part: $ \sqrt{512 m^3} = 16\sqrt{2} \cdot m \cdot \sqrt{m} = 16m\sqrt{2m} \]
The simplified expression is:
16m2m\boxed{16m\sqrt{2m}}

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