Question

determine which expression is equal to $49x^{\frac{3}{2}}$.\
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\\(\bigcirc\\) b) $7x\sqrt{x}$\
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\\(\bigcirc\\) d) $343x^{\frac{1}{2}}$\
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\\(\bigcirc\\) c) $49x\sqrt{x}$\
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\\(\bigcirc\\) a) $7x^{\frac{1}{2}}$

Explanation:

Step1: Recall exponent rule

Recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) and \(x^{\frac{3}{2}} = x^{1+\frac{1}{2}}=x\times x^{\frac{1}{2}}=x\sqrt{x}\) (since \(x^{\frac{1}{2}}=\sqrt{x}\)). Also, \(49 = 7^2\), but we can directly analyze the coefficient and the variable part.

Step2: Analyze the given expression

The given expression is \(49x^{\frac{3}{2}}\). Using the exponent rule for \(x^{\frac{3}{2}}\), we can rewrite \(x^{\frac{3}{2}}\) as \(x\sqrt{x}\) (because \(x^{\frac{3}{2}}=x^{1 + \frac{1}{2}}=x^1\times x^{\frac{1}{2}}=x\sqrt{x}\)). Then the expression \(49x^{\frac{3}{2}}\) becomes \(49\times x\sqrt{x}=49x\sqrt{x}\).

Step3: Check the options

• Option a: \(7x^{\frac{1}{2}}=7\sqrt{x}\), not equal to \(49x^{\frac{3}{2}}\).
• Option b: \(7x\sqrt{x}\), coefficient is 7, but our coefficient is 49, so not equal.
• Option c: \(49x\sqrt{x}\), which matches our rewritten form of \(49x^{\frac{3}{2}}\).
• Option d: \(343x^{\frac{1}{3}}\), the exponent and coefficient are different, so not equal.

Answer:

c) \(49x\sqrt{x}\)