Solved on Jan 24, 2024

Simplify $\sqrt{24 p^{13} q^{6}}$ , assuming all variables are positive. Express exponents as positive values.

STEP 1

Assumptions
1. All variables are positive.
2. The expression to be simplified is $\sqrt{24 p^{13} q^{6}}$ .
3. All exponents should be positive in the simplified form.

STEP 2

First, we need to break down the number 24 into its prime factors.
$24 = 2^3 \cdot 3$

STEP 3

Now, we rewrite the expression with the prime factorization of 24.
$\sqrt{24 p^{13} q^{6}} = \sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}}$

STEP 4

Next, we look for perfect squares within the radicand (the expression inside the square root) that we can take out of the square root. A perfect square is a number or a variable raised to an even power.

STEP 5

We identify the perfect squares in the radicand:
1. $2^3$ can be written as $2^2 \cdot 2$ where $2^2$ is a perfect square.
2. $p^{13}$ can be written as $p^{12} \cdot p$ where $p^{12}$ is a perfect square because 12 is an even number.
3. $q^{6}$ is already a perfect square because 6 is an even number.

STEP 6

Rewrite the radicand by separating the perfect squares from the non-perfect squares.
$\sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}} = \sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}}$

STEP 7

Now, take the square root of the perfect squares, remembering that the square root of a product is the product of the square roots.
$\sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}} = \sqrt{2^2} \cdot \sqrt{p^{12}} \cdot \sqrt{q^{6}} \cdot \sqrt{2 \cdot 3 \cdot p}$

STEP 8

Calculate the square roots of the perfect squares.
$\sqrt{2^2} = 2$ $\sqrt{p^{12}} = p^6$ $\sqrt{q^{6}} = q^3$

STEP 9

Combine the results from the previous step with the remaining radicand.
$\sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}} = 2 \cdot p^6 \cdot q^3 \cdot \sqrt{2 \cdot 3 \cdot p}$

STEP 10

Simplify the remaining radicand by multiplying the numbers under the square root.
$\sqrt{2 \cdot 3 \cdot p} = \sqrt{6p}$

STEP 11

Combine the simplified square root with the numbers and variables taken out of the original square root.
$2 \cdot p^6 \cdot q^3 \cdot \sqrt{6p}$

STEP 12

The expression is now simplified with all exponents positive.
$\sqrt{24 p^{13} q^{6}} = 2 p^6 q^3 \sqrt{6p}$
This is the simplified form of the given expression.

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Simplify $\sqrt{24 p^{13} q^{6}}$ , assuming all variables are positive. Express exponents as positive values.

STEP 1

Assumptions
1. All variables are positive.
2. The expression to be simplified is $\sqrt{24 p^{13} q^{6}}$ .
3. All exponents should be positive in the simplified form.

STEP 2

First, we need to break down the number 24 into its prime factors.
$24 = 2^3 \cdot 3$

STEP 3

Now, we rewrite the expression with the prime factorization of 24.
$\sqrt{24 p^{13} q^{6}} = \sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}}$

STEP 4

Next, we look for perfect squares within the radicand (the expression inside the square root) that we can take out of the square root. A perfect square is a number or a variable raised to an even power.

STEP 5

STEP 6

Rewrite the radicand by separating the perfect squares from the non-perfect squares.
$\sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}} = \sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}}$

STEP 7

Now, take the square root of the perfect squares, remembering that the square root of a product is the product of the square roots.
$\sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}} = \sqrt{2^2} \cdot \sqrt{p^{12}} \cdot \sqrt{q^{6}} \cdot \sqrt{2 \cdot 3 \cdot p}$

STEP 8

Calculate the square roots of the perfect squares.
$\sqrt{2^2} = 2$ $\sqrt{p^{12}} = p^6$ $\sqrt{q^{6}} = q^3$

STEP 9

Combine the results from the previous step with the remaining radicand.
$\sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}} = 2 \cdot p^6 \cdot q^3 \cdot \sqrt{2 \cdot 3 \cdot p}$

STEP 10

Simplify the remaining radicand by multiplying the numbers under the square root.
$\sqrt{2 \cdot 3 \cdot p} = \sqrt{6p}$

STEP 11

Combine the simplified square root with the numbers and variables taken out of the original square root.
$2 \cdot p^6 \cdot q^3 \cdot \sqrt{6p}$

STEP 12

The expression is now simplified with all exponents positive.
$\sqrt{24 p^{13} q^{6}} = 2 p^6 q^3 \sqrt{6p}$
This is the simplified form of the given expression.

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Simplify 24p13q6\sqrt{24 p^{13} q^{6}}24p13q6​, assuming all variables are positive. Express exponents as positive values.

STEP 1

Assumptions1. All variables are positive.2. The expression to be simplified is 24p13q6\sqrt{24 p^{13} q^{6}}24p13q6​.3. All exponents should be positive in the simplified form.

STEP 2

First, we need to break down the number 24 into its prime factors.24=23⋅324 = 2^3 \cdot 324=23⋅3

STEP 3

Now, we rewrite the expression with the prime factorization of 24.24p13q6=23⋅3⋅p13⋅q6\sqrt{24 p^{13} q^{6}} = \sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}}24p13q6​=23⋅3⋅p13⋅q6​

STEP 4

Next, we look for perfect squares within the radicand (the expression inside the square root) that we can take out of the square root. A perfect square is a number or a variable raised to an even power.

STEP 5

STEP 6

Rewrite the radicand by separating the perfect squares from the non-perfect squares.23⋅3⋅p13⋅q6=22⋅2⋅3⋅p12⋅p⋅q6\sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}} = \sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}}23⋅3⋅p13⋅q6​=22⋅2⋅3⋅p12⋅p⋅q6​

STEP 7

STEP 8

Calculate the square roots of the perfect squares.22=2\sqrt{2^2} = 222​=2 p12=p6\sqrt{p^{12}} = p^6p12​=p6 q6=q3\sqrt{q^{6}} = q^3q6​=q3

STEP 9

Combine the results from the previous step with the remaining radicand.22⋅2⋅3⋅p12⋅p⋅q6=2⋅p6⋅q3⋅2⋅3⋅p\sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}} = 2 \cdot p^6 \cdot q^3 \cdot \sqrt{2 \cdot 3 \cdot p}22⋅2⋅3⋅p12⋅p⋅q6​=2⋅p6⋅q3⋅2⋅3⋅p​

STEP 10

Simplify the remaining radicand by multiplying the numbers under the square root.2⋅3⋅p=6p\sqrt{2 \cdot 3 \cdot p} = \sqrt{6p}2⋅3⋅p​=6p​

STEP 11

Combine the simplified square root with the numbers and variables taken out of the original square root.2⋅p6⋅q3⋅6p2 \cdot p^6 \cdot q^3 \cdot \sqrt{6p}2⋅p6⋅q3⋅6p​

STEP 12

The expression is now simplified with all exponents positive.24p13q6=2p6q36p\sqrt{24 p^{13} q^{6}} = 2 p^6 q^3 \sqrt{6p}24p13q6​=2p6q36p​This is the simplified form of the given expression.

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Simplify $\sqrt{24 p^{13} q^{6}}$ , assuming all variables are positive. Express exponents as positive values.

Assumptions
1. All variables are positive.
2. The expression to be simplified is $\sqrt{24 p^{13} q^{6}}$ .
3. All exponents should be positive in the simplified form.

First, we need to break down the number 24 into its prime factors.
$24 = 2^3 \cdot 3$

Now, we rewrite the expression with the prime factorization of 24.
$\sqrt{24 p^{13} q^{6}} = \sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}}$

Rewrite the radicand by separating the perfect squares from the non-perfect squares.
$\sqrt{2^3 \cdot 3 \cdot p^{13} \cdot q^{6}} = \sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}}$

Calculate the square roots of the perfect squares.
$\sqrt{2^2} = 2$ $\sqrt{p^{12}} = p^6$ $\sqrt{q^{6}} = q^3$

Combine the results from the previous step with the remaining radicand.
$\sqrt{2^2 \cdot 2 \cdot 3 \cdot p^{12} \cdot p \cdot q^{6}} = 2 \cdot p^6 \cdot q^3 \cdot \sqrt{2 \cdot 3 \cdot p}$

Simplify the remaining radicand by multiplying the numbers under the square root.
$\sqrt{2 \cdot 3 \cdot p} = \sqrt{6p}$

Combine the simplified square root with the numbers and variables taken out of the original square root.
$2 \cdot p^6 \cdot q^3 \cdot \sqrt{6p}$

The expression is now simplified with all exponents positive.
$\sqrt{24 p^{13} q^{6}} = 2 p^6 q^3 \sqrt{6p}$
This is the simplified form of the given expression.