Solved on Mar 05, 2024

Simplify the expression $8 \sqrt{24 x^{4}}+9 \sqrt{54 x^{4}}$ by combining like radicals.

STEP 1

Assumptions
1. The variable $x$ represents a nonnegative real number.
2. The operation to be performed is the addition of two radical expressions.
3. Like radicals can be combined by adding their coefficients.

STEP 2

First, we need to simplify the radicals by factoring the radicands into perfect squares and other factors.
$\sqrt{24x^4} = \sqrt{4 \cdot 6 \cdot x^4}$ $\sqrt{54x^4} = \sqrt{9 \cdot 6 \cdot x^4}$

STEP 3

Since $x^4$ is a perfect square (as $x$ is nonnegative), and $4$ and $9$ are also perfect squares, we can take them out of the square root.
$\sqrt{4 \cdot 6 \cdot x^4} = \sqrt{4} \cdot \sqrt{6} \cdot \sqrt{x^4}$ $\sqrt{9 \cdot 6 \cdot x^4} = \sqrt{9} \cdot \sqrt{6} \cdot \sqrt{x^4}$

STEP 4

Calculate the square roots of the perfect squares.
$\sqrt{4} = 2$ $\sqrt{9} = 3$ $\sqrt{x^4} = x^2$

STEP 5

Substitute the square roots of the perfect squares back into the expressions.
$\sqrt{24x^4} = 2 \cdot \sqrt{6} \cdot x^2$ $\sqrt{54x^4} = 3 \cdot \sqrt{6} \cdot x^2$

STEP 6

Now, rewrite the original expression using the simplified radicals.
$8 \sqrt{24 x^{4}} + 9 \sqrt{54 x^{4}} = 8(2 \cdot \sqrt{6} \cdot x^2) + 9(3 \cdot \sqrt{6} \cdot x^2)$

STEP 7

Multiply the coefficients inside the parentheses.
$8(2 \cdot \sqrt{6} \cdot x^2) = 16 \cdot \sqrt{6} \cdot x^2$ $9(3 \cdot \sqrt{6} \cdot x^2) = 27 \cdot \sqrt{6} \cdot x^2$

STEP 8

Combine the like radicals by adding their coefficients.
$16 \cdot \sqrt{6} \cdot x^2 + 27 \cdot \sqrt{6} \cdot x^2 = (16 + 27) \cdot \sqrt{6} \cdot x^2$

STEP 9

Add the coefficients.
$16 + 27 = 43$

STEP 10

Write the final simplified expression.
$43 \cdot \sqrt{6} \cdot x^2$
The simplified expression is $43 \sqrt{6} x^2$ .

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Simplify the expression $8 \sqrt{24 x^{4}}+9 \sqrt{54 x^{4}}$ by combining like radicals.

STEP 1

Assumptions
1. The variable $x$ represents a nonnegative real number.
2. The operation to be performed is the addition of two radical expressions.
3. Like radicals can be combined by adding their coefficients.

STEP 2

First, we need to simplify the radicals by factoring the radicands into perfect squares and other factors.
$\sqrt{24x^4} = \sqrt{4 \cdot 6 \cdot x^4}$ $\sqrt{54x^4} = \sqrt{9 \cdot 6 \cdot x^4}$

STEP 3

Since $x^4$ is a perfect square (as $x$ is nonnegative), and $4$ and $9$ are also perfect squares, we can take them out of the square root.
$\sqrt{4 \cdot 6 \cdot x^4} = \sqrt{4} \cdot \sqrt{6} \cdot \sqrt{x^4}$ $\sqrt{9 \cdot 6 \cdot x^4} = \sqrt{9} \cdot \sqrt{6} \cdot \sqrt{x^4}$

STEP 4

Calculate the square roots of the perfect squares.
$\sqrt{4} = 2$ $\sqrt{9} = 3$ $\sqrt{x^4} = x^2$

STEP 5

Substitute the square roots of the perfect squares back into the expressions.
$\sqrt{24x^4} = 2 \cdot \sqrt{6} \cdot x^2$ $\sqrt{54x^4} = 3 \cdot \sqrt{6} \cdot x^2$

STEP 6

Now, rewrite the original expression using the simplified radicals.
$8 \sqrt{24 x^{4}} + 9 \sqrt{54 x^{4}} = 8(2 \cdot \sqrt{6} \cdot x^2) + 9(3 \cdot \sqrt{6} \cdot x^2)$

STEP 7

Multiply the coefficients inside the parentheses.
$8(2 \cdot \sqrt{6} \cdot x^2) = 16 \cdot \sqrt{6} \cdot x^2$ $9(3 \cdot \sqrt{6} \cdot x^2) = 27 \cdot \sqrt{6} \cdot x^2$

STEP 8

Combine the like radicals by adding their coefficients.
$16 \cdot \sqrt{6} \cdot x^2 + 27 \cdot \sqrt{6} \cdot x^2 = (16 + 27) \cdot \sqrt{6} \cdot x^2$

STEP 9

Add the coefficients.
$16 + 27 = 43$

STEP 10

Write the final simplified expression.
$43 \cdot \sqrt{6} \cdot x^2$
The simplified expression is $43 \sqrt{6} x^2$ .

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Simplify the expression 824x4+954x48 \sqrt{24 x^{4}}+9 \sqrt{54 x^{4}}824x4​+954x4​ by combining like radicals.

STEP 1

Assumptions1. The variable xxx represents a nonnegative real number.2. The operation to be performed is the addition of two radical expressions.3. Like radicals can be combined by adding their coefficients.

STEP 2

First, we need to simplify the radicals by factoring the radicands into perfect squares and other factors.24x4=4⋅6⋅x4\sqrt{24x^4} = \sqrt{4 \cdot 6 \cdot x^4}24x4​=4⋅6⋅x4​ 54x4=9⋅6⋅x4\sqrt{54x^4} = \sqrt{9 \cdot 6 \cdot x^4}54x4​=9⋅6⋅x4​

STEP 3

STEP 4

Calculate the square roots of the perfect squares.4=2\sqrt{4} = 24​=2 9=3\sqrt{9} = 39​=3 x4=x2\sqrt{x^4} = x^2x4​=x2

STEP 5

Substitute the square roots of the perfect squares back into the expressions.24x4=2⋅6⋅x2\sqrt{24x^4} = 2 \cdot \sqrt{6} \cdot x^224x4​=2⋅6​⋅x2 54x4=3⋅6⋅x2\sqrt{54x^4} = 3 \cdot \sqrt{6} \cdot x^254x4​=3⋅6​⋅x2

STEP 6

Now, rewrite the original expression using the simplified radicals.824x4+954x4=8(2⋅6⋅x2)+9(3⋅6⋅x2)8 \sqrt{24 x^{4}} + 9 \sqrt{54 x^{4}} = 8(2 \cdot \sqrt{6} \cdot x^2) + 9(3 \cdot \sqrt{6} \cdot x^2)824x4​+954x4​=8(2⋅6​⋅x2)+9(3⋅6​⋅x2)

STEP 7

Multiply the coefficients inside the parentheses.8(2⋅6⋅x2)=16⋅6⋅x28(2 \cdot \sqrt{6} \cdot x^2) = 16 \cdot \sqrt{6} \cdot x^28(2⋅6​⋅x2)=16⋅6​⋅x2 9(3⋅6⋅x2)=27⋅6⋅x29(3 \cdot \sqrt{6} \cdot x^2) = 27 \cdot \sqrt{6} \cdot x^29(3⋅6​⋅x2)=27⋅6​⋅x2

STEP 8

Combine the like radicals by adding their coefficients.16⋅6⋅x2+27⋅6⋅x2=(16+27)⋅6⋅x216 \cdot \sqrt{6} \cdot x^2 + 27 \cdot \sqrt{6} \cdot x^2 = (16 + 27) \cdot \sqrt{6} \cdot x^216⋅6​⋅x2+27⋅6​⋅x2=(16+27)⋅6​⋅x2

STEP 9

Add the coefficients.16+27=4316 + 27 = 4316+27=43

STEP 10

Write the final simplified expression.43⋅6⋅x243 \cdot \sqrt{6} \cdot x^243⋅6​⋅x2The simplified expression is 436x243 \sqrt{6} x^2436​x2.

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Simplify the expression $8 \sqrt{24 x^{4}}+9 \sqrt{54 x^{4}}$ by combining like radicals.

Assumptions
1. The variable $x$ represents a nonnegative real number.
2. The operation to be performed is the addition of two radical expressions.
3. Like radicals can be combined by adding their coefficients.

First, we need to simplify the radicals by factoring the radicands into perfect squares and other factors.
$\sqrt{24x^4} = \sqrt{4 \cdot 6 \cdot x^4}$ $\sqrt{54x^4} = \sqrt{9 \cdot 6 \cdot x^4}$

Calculate the square roots of the perfect squares.
$\sqrt{4} = 2$ $\sqrt{9} = 3$ $\sqrt{x^4} = x^2$

Substitute the square roots of the perfect squares back into the expressions.
$\sqrt{24x^4} = 2 \cdot \sqrt{6} \cdot x^2$ $\sqrt{54x^4} = 3 \cdot \sqrt{6} \cdot x^2$

Now, rewrite the original expression using the simplified radicals.
$8 \sqrt{24 x^{4}} + 9 \sqrt{54 x^{4}} = 8(2 \cdot \sqrt{6} \cdot x^2) + 9(3 \cdot \sqrt{6} \cdot x^2)$

Multiply the coefficients inside the parentheses.
$8(2 \cdot \sqrt{6} \cdot x^2) = 16 \cdot \sqrt{6} \cdot x^2$ $9(3 \cdot \sqrt{6} \cdot x^2) = 27 \cdot \sqrt{6} \cdot x^2$

Combine the like radicals by adding their coefficients.
$16 \cdot \sqrt{6} \cdot x^2 + 27 \cdot \sqrt{6} \cdot x^2 = (16 + 27) \cdot \sqrt{6} \cdot x^2$

Add the coefficients.
$16 + 27 = 43$

Write the final simplified expression.
$43 \cdot \sqrt{6} \cdot x^2$
The simplified expression is $43 \sqrt{6} x^2$ .