Question

determine which expression is equal to $49x^{\frac{1}{2}}$.
a) $7x^4$
b) $7x\sqrt{x}$
c) $49x\sqrt{x}$
d) $343x^{\frac{1}{2}}$

Explanation:

Step1: Rewrite radical as exponent

Recall that $\sqrt{x} = x^{\frac{1}{2}}$.

Step2: Convert each option to exponent form

Option a:

$7x^{\frac{1}{2}}
eq 49x^{\frac{1}{2}}$

Option b:

$7x\sqrt{x} = 7x \cdot x^{\frac{1}{2}} = 7x^{1+\frac{1}{2}} = 7x^{\frac{3}{2}}
eq 49x^{\frac{1}{2}}$

Option c:

$49x\sqrt{x} = 49x \cdot x^{\frac{1}{2}} = 49x^{1+\frac{1}{2}} = 49x^{\frac{3}{2}}
eq 49x^{\frac{1}{2}}$

Option d:

$343x^{\frac{1}{2}}
eq 49x^{\frac{1}{2}}$

*Note: Correcting a potential typo in the original question: if the target was $49x^{\frac{3}{2}}$, option c would be correct. Assuming the target is $49x^{\frac{1}{2}}$, none match, but re-evaluating standard radical conversions:
Wait, recheck the question: if the original expression is $\sqrt{49x}$, that equals $7\sqrt{x}=7x^{\frac{1}{2}}$, but the question states $49x^{\frac{1}{2}}$. If we assume a misprint and the option was $49\sqrt{x}$, that would match, but given the options, if the target is $49x^{\frac{3}{2}}$, then:
$49x\sqrt{x}=49x \cdot x^{\frac{1}{2}}=49x^{\frac{3}{2}}$, which matches. Given common textbook problems, this is likely the intended target.*

Revised Step2 (assuming target is $49x^{\frac{3}{2}}$):

Option c:

$49x\sqrt{x} = 49x \cdot x^{\frac{1}{2}} = 49x^{\frac{3}{2}}$

Answer:

c) $49x\sqrt{x}$