To find the equivalent expression for \( x^{16} \) using the properties of exponents and the expressions provided, let's first analyze the options you mentioned. However, it seems there may have been a formatting issue with your request. I'll explain how to work with rational exponents and square roots, then you can use these properties for the specific expressions you have.
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Understanding Rational Exponents:
- The expression \( \sqrt{x} \) can be rewritten as \( x^{1/2} \).
- The square root of an expression can be expressed using an exponent as \( x^{m/n} \), where \( m \) is the power inside the root and \( n \) is the type of root (for square roots, \( n = 2 \)).
- Thus, \( \sqrt{x^n} = x^{n/2} \).
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Using Properties of Exponents:
- When multiplying expressions with the same base, you can add the exponents: \( x^a \cdot x^b = x^{a+b} \).
- When dividing, you subtract the exponents: \( x^a / x^b = x^{a-b} \).
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Equivalent Expressions: To simplify the expressions:
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For the expression \( 3\sqrt{x} \): \[ 3\sqrt{x} = 3x^{1/2} \]
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For the expression \( 36\sqrt{x} \): \[ 36\sqrt{x} = 36x^{1/2} \]
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For the expression \( x^{1/3}\sqrt{x} \): \[ x^{1/3}\sqrt{x} = x^{1/3} \cdot x^{1/2} = x^{(1/3 + 1/2)} = x^{(2/6 + 3/6)} = x^{5/6} \]
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For the expression \( \sqrt{x} \): \[ \sqrt{x} = x^{1/2} \]
So, none of these options seem to directly equal \( x^{16} \), given the simplifications above. If you want an equivalent expression for \( x^{16} \) in terms of square roots or rational exponents, you may consider:
- \( x^{16} = \sqrt{x^{32}} \)
- \( x^{16} = \sqrt[4]{x^{64}} \)
- \( x^{16} = (x^8)^2 \)
However, since none of the given expressions directly match, please check if there are additional options or clarify the expressions further, and I'd be happy to assist you!
5 answers