To simplify the expression \( \frac{\sqrt{8}}{\sqrt{2}} \) using properties of exponents, we'll break down the steps.
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Rewrite the square roots using exponents: \[ \sqrt{8} = 8^{1/2} \quad \text{and} \quad \sqrt{2} = 2^{1/2} \] So the expression can be rewritten as: \[ \frac{\sqrt{8}}{\sqrt{2}} = \frac{8^{1/2}}{2^{1/2}} \]
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Express \(8\) in terms of \(2\): Since \(8 = 2^3\), we can replace \(8\) in our expression: \[ 8^{1/2} = (2^3)^{1/2} = 2^{3/2} \]
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Substitute: Now substitute \(8^{1/2}\) into our expression: \[ \frac{8^{1/2}}{2^{1/2}} = \frac{2^{3/2}}{2^{1/2}} \]
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Use the properties of exponents to simplify: When dividing exponents with the same base, we subtract the exponents: \[ 2^{3/2 - 1/2} = 2^{(3/2 - 1/2)} = 2^{2/2} = 2^1 = 2 \]
So, \[ \frac{\sqrt{8}}{\sqrt{2}} = 2 \]
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Finding the cube root of \(8\): The cube root of \(8\) is: \[ \sqrt[3]{8} = 2 \]
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Rewriting rational exponents as radicals: To rewrite a rational exponent \(x^{m/n}\) as a radical, it can be represented as: \[ x^{m/n} = \sqrt[n]{x^m} \] For example:
- \(8^{1/3} = \sqrt[3]{8}\)
Both rational exponent formats and their corresponding radical forms are equivalent, and we have simplified the main expression \( \frac{\sqrt{8}}{\sqrt{2}} \) to \(2\) and found the cube root of \(8\) also to be \(2\).
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