Simplify the Radical Expression: √(49x²)/x

Express the following root as a power:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Remember that in a square root (also called "root to the power of 2") we don't write the root's power as shown

$n=2$

Meaning:

$\sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}$

Let's return to the problem and use the root definition that we mentioned above to convert the root in the fraction's numerator:

$\frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}$

Remember the two following laws of exponents:

a. The law of exponents for a power applied to a product inside of parentheses:

$(a\cdot b)^n=a^n\cdot b^n$

b. The law of exponents for a power of a power:

$(a^m)^n=a^{m\cdot n}$

Let's apply these laws to the fraction's numerator in the expression that we obtained in the last step:

$\frac{(49x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}\cdot(x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x}$

$$In the first stage we applied the above-mentioned law of exponents noted in a' and then proceeded to applythe power to both factors of the product (in parentheses) in the fraction's numerator. We we careful to use parentheses given that one of the factors in the parentheses is already raised to a power.

In the second stage we applied the second law of exponents mentioned in b' to the second factor in the product,

Let's simplify the expression that we obtained:

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

In the first stage we converted the fraction's power back to a root, for the first factor in the product, using the definition of root as a power mentioned at the beginning of the solution ( in the opposite direction)

Additionally- we calculated the product in the exponent of the second factor in the product in the fraction's numerator in the expression that we obtained. We then we simplified the resulting fraction.

Finish the calculation and proceed to simplify the resulting fraction:

$\frac{7x^1}{x}=\frac{7\not{x}}{\not{x}}=7$

Let's summarize the various steps of the solution that we obtained thus far, as shown below:

$\frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}=\frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x} =7$

Therefore the correct answer is answer c.