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

Question

Solve the following exercise:

49x2x= \frac{\sqrt{49x^2}}{x}=

Video Solution

Solution Steps

00:00 Solve
00:03 when there is a root of multiplied terms (A times B)
00:06 we can break it down to root (A) times root (B)
00:10 we will use this formula in our exercise
00:21 the root of any number (A) squared cancels out the square
00:25 we will use this formula in our exercise, and cancel out the square:
00:30 let's simplify what we can
00:36 we'll break down 49 to 7 squared
00:39 root cancels square
00:42 and this is the solution to the question

Step-by-Step Solution

Let's use the definition of root as a power:

an=a1n \sqrt[n]{a}=a^{\frac{1}{n}}

when we remember that in a square root (also called "root to the power of 2") we don't write the root's power and:

n=2 n=2

meaning:

a=a2=a12 \sqrt{a}=\sqrt[2]{a}=a^{\frac{1}{2}}

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

49x2x=(49x2)12x \frac{\sqrt{49x^2}}{x}=\frac{(49x^2)^{\frac{1}{2}}}{x}

Now let's recall two laws of exponents:

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

(ab)n=anbn (a\cdot b)^n=a^n\cdot b^n

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

(am)n=amn (a^m)^n=a^{m\cdot n}

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

(49x2)12x=4912(x2)12x=4912x212x \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}

where in the first stage we applied the above-mentioned law of exponents noted in a' and applied the power to both factors of the product in parentheses in the fraction's numerator, we did this carefully using parentheses since 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 we got:

4912x212x=49x22x=7x1x \frac{49^{\frac{1}{2}}x^{2\cdot\frac{1}{2}}}{x}=\frac{\sqrt{49}x^{\frac{2}{2}}}{x}=\frac{7x^1}{x}

where in the first stage we converted back the fraction's power 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, but 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 we got, then we simplified the resulting fraction in that exponent for that factor.

Let's finish the calculation and simplify the resulting fraction:

7x1x=7=7 \frac{7x^1}{x}=\frac{7\not{x}}{\not{x}}=7

Let's summarize the solution steps so far, we got that:

49x2x=(49x2)12x=4912x212x=7 \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.

Answer

7 7