Solved on Feb 23, 2024

Simplify the quotient $\sqrt{16 x^{3}} \div \sqrt{8 x}$ when $x>0$ .

STEP 1

Assumptions
1. We are given the expression $\sqrt{16 x^{3}} \div \sqrt{8 x}$ .
2. We are looking for an equivalent expression.
3. $x > 0$ ensures that all radicals are defined.

STEP 2

First, we can simplify the expression by combining the radicals into one, since the division of two radicals can be expressed as a single radical.
$\frac{\sqrt{16 x^{3}}}{\sqrt{8 x}} = \sqrt{\frac{16 x^{3}}{8 x}}$

STEP 3

Next, we simplify the fraction inside the radical by dividing both the numerator and the denominator by $8x$ .
$\sqrt{\frac{16 x^{3}}{8 x}} = \sqrt{\frac{2 \cdot 8 x^{3}}{8 x}}$

STEP 4

We can cancel out the common factor of $8x$ in the numerator and the denominator.
$\sqrt{\frac{2 \cdot 8 x^{3}}{8 x}} = \sqrt{2 \cdot x^{2}}$

STEP 5

Now, we can separate the radical into two parts, one for the constant and one for the variable.
$\sqrt{2 \cdot x^{2}} = \sqrt{2} \cdot \sqrt{x^{2}}$

STEP 6

Since $x > 0$ , we can simplify $\sqrt{x^{2}}$ to $x$ .
$\sqrt{2} \cdot \sqrt{x^{2}} = \sqrt{2} \cdot x$

STEP 7

Finally, we rewrite the expression in a more conventional form by placing the variable before the radical.
$\sqrt{2} \cdot x = x \sqrt{2}$
The equivalent expression to the given quotient is $x \sqrt{2}$ , which corresponds to option D.

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Simplify the quotient $\sqrt{16 x^{3}} \div \sqrt{8 x}$ when $x>0$ .

STEP 1

Assumptions
1. We are given the expression $\sqrt{16 x^{3}} \div \sqrt{8 x}$ .
2. We are looking for an equivalent expression.
3. $x > 0$ ensures that all radicals are defined.

STEP 2

First, we can simplify the expression by combining the radicals into one, since the division of two radicals can be expressed as a single radical.
$\frac{\sqrt{16 x^{3}}}{\sqrt{8 x}} = \sqrt{\frac{16 x^{3}}{8 x}}$

STEP 3

Next, we simplify the fraction inside the radical by dividing both the numerator and the denominator by $8x$ .
$\sqrt{\frac{16 x^{3}}{8 x}} = \sqrt{\frac{2 \cdot 8 x^{3}}{8 x}}$

STEP 4

We can cancel out the common factor of $8x$ in the numerator and the denominator.
$\sqrt{\frac{2 \cdot 8 x^{3}}{8 x}} = \sqrt{2 \cdot x^{2}}$

STEP 5

Now, we can separate the radical into two parts, one for the constant and one for the variable.
$\sqrt{2 \cdot x^{2}} = \sqrt{2} \cdot \sqrt{x^{2}}$

STEP 6

Since $x > 0$ , we can simplify $\sqrt{x^{2}}$ to $x$ .
$\sqrt{2} \cdot \sqrt{x^{2}} = \sqrt{2} \cdot x$

STEP 7

Finally, we rewrite the expression in a more conventional form by placing the variable before the radical.
$\sqrt{2} \cdot x = x \sqrt{2}$
The equivalent expression to the given quotient is $x \sqrt{2}$ , which corresponds to option D.

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Simplify the quotient 16x3÷8x\sqrt{16 x^{3}} \div \sqrt{8 x}16x3​÷8x​ when x>0x>0x>0.

STEP 1

Assumptions1. We are given the expression 16x3÷8x\sqrt{16 x^{3}} \div \sqrt{8 x}16x3​÷8x​.2. We are looking for an equivalent expression.3. x>0x > 0x>0 ensures that all radicals are defined.

STEP 2

First, we can simplify the expression by combining the radicals into one, since the division of two radicals can be expressed as a single radical.16x38x=16x38x\frac{\sqrt{16 x^{3}}}{\sqrt{8 x}} = \sqrt{\frac{16 x^{3}}{8 x}}8x​16x3​​=8x16x3​​

STEP 3

Next, we simplify the fraction inside the radical by dividing both the numerator and the denominator by 8x8x8x.16x38x=2⋅8x38x\sqrt{\frac{16 x^{3}}{8 x}} = \sqrt{\frac{2 \cdot 8 x^{3}}{8 x}}8x16x3​​=8x2⋅8x3​​

STEP 4

We can cancel out the common factor of 8x8x8x in the numerator and the denominator.2⋅8x38x=2⋅x2\sqrt{\frac{2 \cdot 8 x^{3}}{8 x}} = \sqrt{2 \cdot x^{2}}8x2⋅8x3​​=2⋅x2​

STEP 5

Now, we can separate the radical into two parts, one for the constant and one for the variable.2⋅x2=2⋅x2\sqrt{2 \cdot x^{2}} = \sqrt{2} \cdot \sqrt{x^{2}}2⋅x2​=2​⋅x2​

STEP 6

Since x>0x > 0x>0, we can simplify x2\sqrt{x^{2}}x2​ to xxx.2⋅x2=2⋅x\sqrt{2} \cdot \sqrt{x^{2}} = \sqrt{2} \cdot x2​⋅x2​=2​⋅x

STEP 7

Finally, we rewrite the expression in a more conventional form by placing the variable before the radical.2⋅x=x2\sqrt{2} \cdot x = x \sqrt{2}2​⋅x=x2​The equivalent expression to the given quotient is x2x \sqrt{2}x2​, which corresponds to option D.

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Simplify the quotient $\sqrt{16 x^{3}} \div \sqrt{8 x}$ when $x>0$ .

Assumptions
1. We are given the expression $\sqrt{16 x^{3}} \div \sqrt{8 x}$ .
2. We are looking for an equivalent expression.
3. $x > 0$ ensures that all radicals are defined.

First, we can simplify the expression by combining the radicals into one, since the division of two radicals can be expressed as a single radical.
$\frac{\sqrt{16 x^{3}}}{\sqrt{8 x}} = \sqrt{\frac{16 x^{3}}{8 x}}$

Next, we simplify the fraction inside the radical by dividing both the numerator and the denominator by $8x$ .
$\sqrt{\frac{16 x^{3}}{8 x}} = \sqrt{\frac{2 \cdot 8 x^{3}}{8 x}}$

We can cancel out the common factor of $8x$ in the numerator and the denominator.
$\sqrt{\frac{2 \cdot 8 x^{3}}{8 x}} = \sqrt{2 \cdot x^{2}}$

Now, we can separate the radical into two parts, one for the constant and one for the variable.
$\sqrt{2 \cdot x^{2}} = \sqrt{2} \cdot \sqrt{x^{2}}$

Since $x > 0$ , we can simplify $\sqrt{x^{2}}$ to $x$ .
$\sqrt{2} \cdot \sqrt{x^{2}} = \sqrt{2} \cdot x$

Finally, we rewrite the expression in a more conventional form by placing the variable before the radical.
$\sqrt{2} \cdot x = x \sqrt{2}$
The equivalent expression to the given quotient is $x \sqrt{2}$ , which corresponds to option D.