Math / Algebra

QuestionSimplify $\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{4}}}$ a. $\frac{d^{d^{2}}}{2 \sqrt{c}}$ c. $\frac{d^{3}}{2-c}$ b. $\frac{d^{5} \sqrt{c}}{2 c}$ d. $\frac{d^{2}}{4 c}$

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with square roots and variables. Watch out! Remember your exponent rules and don't forget to simplify the square roots completely!

STEP 2

1. Combine the square roots
2. Simplify the expression

STEP 3

We can rewrite the expression as a single square root to make it easier to work with.
Remember, $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ .
So, we have: $\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{-4}}} = \sqrt{\frac{c^{2} d^{6}}{4 c^{3} d^{-4}}}$

STEP 4

Now, let's simplify the fraction inside the square root.
Remember when dividing like bases, we subtract the exponents.
We'll divide the $c$ terms and the $d$ terms separately.
For the $c$ terms, we have $c^2$ divided by $c^3$ , which is $c^{2-3} = c^{-1}$ .
For the $d$ terms, we have $d^6$ divided by $d^{-4}$ , which is $d^{6 - (-4)} = d^{6+4} = d^{10}$ .
Don't forget about the 4 in the denominator!
So, we get: $\sqrt{\frac{c^{2} d^{6}}{4 c^{3} d^{-4}}} = \sqrt{\frac{d^{10}}{4c}}$

STEP 5

We can separate the square root of the fraction into a fraction of square roots.
This will help us simplify further! $\sqrt{\frac{d^{10}}{4c}} = \frac{\sqrt{d^{10}}}{\sqrt{4c}}$

STEP 6

Remember, the square root of $x$ is the same as $x$ to the power of $\frac{1}{2}$ .
So, $\sqrt{d^{10}} = (d^{10})^{\frac{1}{2}}$ .
When we raise a power to a power, we multiply the exponents.
So, $(d^{10})^{\frac{1}{2}} = d^{10 \cdot \frac{1}{2}} = d^5$ .
Awesome!

STEP 7

We can rewrite $\sqrt{4c}$ as $\sqrt{4} \cdot \sqrt{c}$ .
Since $\sqrt{4} = 2$ , our denominator becomes $2\sqrt{c}$ .

STEP 8

Combining our simplified numerator and denominator, we get our final simplified expression: $\frac{d^5}{2\sqrt{c}}$

STEP 9

The simplified expression is $\frac{d^5}{2\sqrt{c}}$ , which matches answer choice (b).

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QuestionSimplify $\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{4}}}$ a. $\frac{d^{d^{2}}}{2 \sqrt{c}}$ c. $\frac{d^{3}}{2-c}$ b. $\frac{d^{5} \sqrt{c}}{2 c}$ d. $\frac{d^{2}}{4 c}$

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with square roots and variables. Watch out! Remember your exponent rules and don't forget to simplify the square roots completely!

STEP 2

1. Combine the square roots
2. Simplify the expression

STEP 3

We can rewrite the expression as a single square root to make it easier to work with.
Remember, $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ .
So, we have: $\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{-4}}} = \sqrt{\frac{c^{2} d^{6}}{4 c^{3} d^{-4}}}$

STEP 4

STEP 5

We can separate the square root of the fraction into a fraction of square roots.
This will help us simplify further! $\sqrt{\frac{d^{10}}{4c}} = \frac{\sqrt{d^{10}}}{\sqrt{4c}}$

STEP 6

Remember, the square root of $x$ is the same as $x$ to the power of $\frac{1}{2}$ .
So, $\sqrt{d^{10}} = (d^{10})^{\frac{1}{2}}$ .
When we raise a power to a power, we multiply the exponents.
So, $(d^{10})^{\frac{1}{2}} = d^{10 \cdot \frac{1}{2}} = d^5$ .
Awesome!

STEP 7

We can rewrite $\sqrt{4c}$ as $\sqrt{4} \cdot \sqrt{c}$ .
Since $\sqrt{4} = 2$ , our denominator becomes $2\sqrt{c}$ .

STEP 8

Combining our simplified numerator and denominator, we get our final simplified expression: $\frac{d^5}{2\sqrt{c}}$

STEP 9

The simplified expression is $\frac{d^5}{2\sqrt{c}}$ , which matches answer choice (b).

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QuestionSimplify c2d64c3d4\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{4}}}4c3d4​c2d6​​ a. dd22c\frac{d^{d^{2}}}{2 \sqrt{c}}2c​dd2​ c. d32−c\frac{d^{3}}{2-c}2−cd3​ b. d5c2c\frac{d^{5} \sqrt{c}}{2 c}2cd5c​​ d. d24c\frac{d^{2}}{4 c}4cd2​

Studdy Solution

STEP 1

What is this asking? We need to simplify a fraction with square roots and variables. Watch out! Remember your exponent rules and don't forget to simplify the square roots completely!

STEP 2

1. Combine the square roots2. Simplify the expression

STEP 3

STEP 4

STEP 5

We can separate the square root of the fraction into a fraction of square roots.This will help us simplify further! d104c=d104c \sqrt{\frac{d^{10}}{4c}} = \frac{\sqrt{d^{10}}}{\sqrt{4c}} 4cd10​​=4c​d10​​

STEP 6

STEP 7

We can rewrite 4c\sqrt{4c}4c​ as 4⋅c\sqrt{4} \cdot \sqrt{c}4​⋅c​.Since 4=2\sqrt{4} = 24​=2, our denominator becomes 2c2\sqrt{c}2c​.

STEP 8

Combining our simplified numerator and denominator, we get our **final simplified expression**: d52c \frac{d^5}{2\sqrt{c}} 2c​d5​

STEP 9

The simplified expression is d52c\frac{d^5}{2\sqrt{c}}2c​d5​, which matches answer choice (b).

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Start learning now

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QuestionSimplify $\frac{\sqrt{c^{2} d^{6}}}{\sqrt{4 c^{3} d^{4}}}$ a. $\frac{d^{d^{2}}}{2 \sqrt{c}}$ c. $\frac{d^{3}}{2-c}$ b. $\frac{d^{5} \sqrt{c}}{2 c}$ d. $\frac{d^{2}}{4 c}$

1. Combine the square roots
2. Simplify the expression

We can separate the square root of the fraction into a fraction of square roots.
This will help us simplify further! $\sqrt{\frac{d^{10}}{4c}} = \frac{\sqrt{d^{10}}}{\sqrt{4c}}$

We can rewrite $\sqrt{4c}$ as $\sqrt{4} \cdot \sqrt{c}$ .
Since $\sqrt{4} = 2$ , our denominator becomes $2\sqrt{c}$ .

Combining our simplified numerator and denominator, we get our final simplified expression: $\frac{d^5}{2\sqrt{c}}$

The simplified expression is $\frac{d^5}{2\sqrt{c}}$ , which matches answer choice (b).