Question Simplify $\sqrt[3]{y^{14}}$ by factoring, where $y$ is a positive real number and the radicand does not contain negative quantities raised to even powers.

Studdy Solution

STEP 1

Assumptions
1. The variable $y$ represents a positive real number.
2. The radicand does not contain negative quantities raised to even powers.
3. We are simplifying the expression $\sqrt[3]{y^{14}}$ by factoring.

STEP 2

Identify the index of the radical and the exponent of the variable within the radicand. The index of the radical is 3 (since it is a cube root), and the exponent of the variable $y$ is 14.

STEP 3

To simplify the expression by factoring, we look for the highest power of $y$ that is a multiple of the index of the radical. In this case, we are looking for the highest power of $y$ that is a multiple of 3.

STEP 4

Since $14 = 3 \times 4 + 2$ , the highest power of $y$ that is a multiple of 3 and less than or equal to 14 is $3 \times 4 = 12$ . We can write $y^{14}$ as $y^{12} \cdot y^2$ .

STEP 5

Now we can rewrite the original expression using the factorization from the previous step.
$\sqrt[3]{y^{14}} = \sqrt[3]{y^{12} \cdot y^2}$

STEP 6

We can now separate the radical into two parts using the property of radicals that states $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ .
$\sqrt[3]{y^{12} \cdot y^2} = \sqrt[3]{y^{12}} \cdot \sqrt[3]{y^2}$

STEP 7

Since $y^{12}$ is a perfect cube (because $12$ is a multiple of $3$ ), we can simplify $\sqrt[3]{y^{12}}$ to $y^4$ .
$\sqrt[3]{y^{12}} = y^4$

STEP 8

The expression now simplifies to:
$y^4 \cdot \sqrt[3]{y^2}$

STEP 9

This is the simplified form of the original expression by factoring. The final answer is:
$\sqrt[3]{y^{14}} = y^4 \cdot \sqrt[3]{y^2}$

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Question Simplify $\sqrt[3]{y^{14}}$ by factoring, where $y$ is a positive real number and the radicand does not contain negative quantities raised to even powers.

Studdy Solution

STEP 1

Assumptions
1. The variable $y$ represents a positive real number.
2. The radicand does not contain negative quantities raised to even powers.
3. We are simplifying the expression $\sqrt[3]{y^{14}}$ by factoring.

STEP 2

Identify the index of the radical and the exponent of the variable within the radicand. The index of the radical is 3 (since it is a cube root), and the exponent of the variable $y$ is 14.

STEP 3

To simplify the expression by factoring, we look for the highest power of $y$ that is a multiple of the index of the radical. In this case, we are looking for the highest power of $y$ that is a multiple of 3.

STEP 4

Since $14 = 3 \times 4 + 2$ , the highest power of $y$ that is a multiple of 3 and less than or equal to 14 is $3 \times 4 = 12$ . We can write $y^{14}$ as $y^{12} \cdot y^2$ .

STEP 5

Now we can rewrite the original expression using the factorization from the previous step.
$\sqrt[3]{y^{14}} = \sqrt[3]{y^{12} \cdot y^2}$

STEP 6

We can now separate the radical into two parts using the property of radicals that states $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ .
$\sqrt[3]{y^{12} \cdot y^2} = \sqrt[3]{y^{12}} \cdot \sqrt[3]{y^2}$

STEP 7

Since $y^{12}$ is a perfect cube (because $12$ is a multiple of $3$ ), we can simplify $\sqrt[3]{y^{12}}$ to $y^4$ .
$\sqrt[3]{y^{12}} = y^4$

STEP 8

The expression now simplifies to:
$y^4 \cdot \sqrt[3]{y^2}$

STEP 9

This is the simplified form of the original expression by factoring. The final answer is:
$\sqrt[3]{y^{14}} = y^4 \cdot \sqrt[3]{y^2}$

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Question Simplify y143\sqrt[3]{y^{14}}3y14​ by factoring, where yyy is a positive real number and the radicand does not contain negative quantities raised to even powers.

Studdy Solution

STEP 1

Assumptions1. The variable yyy represents a positive real number.2. The radicand does not contain negative quantities raised to even powers.3. We are simplifying the expression y143\sqrt[3]{y^{14}}3y14​ by factoring.

STEP 2

Identify the index of the radical and the exponent of the variable within the radicand. The index of the radical is 3 (since it is a cube root), and the exponent of the variable yyy is 14.

STEP 3

To simplify the expression by factoring, we look for the highest power of yyy that is a multiple of the index of the radical. In this case, we are looking for the highest power of yyy that is a multiple of 3.

STEP 4

Since 14=3×4+214 = 3 \times 4 + 214=3×4+2, the highest power of yyy that is a multiple of 3 and less than or equal to 14 is 3×4=123 \times 4 = 123×4=12. We can write y14y^{14}y14 as y12⋅y2y^{12} \cdot y^2y12⋅y2.

STEP 5

Now we can rewrite the original expression using the factorization from the previous step.y143=y12⋅y23 \sqrt[3]{y^{14}} = \sqrt[3]{y^{12} \cdot y^2} 3y14​=3y12⋅y2​

STEP 6

We can now separate the radical into two parts using the property of radicals that states abn=an⋅bn\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}nab​=na​⋅nb​.y12⋅y23=y123⋅y23 \sqrt[3]{y^{12} \cdot y^2} = \sqrt[3]{y^{12}} \cdot \sqrt[3]{y^2} 3y12⋅y2​=3y12​⋅3y2​

STEP 7

Since y12y^{12}y12 is a perfect cube (because 121212 is a multiple of 333), we can simplify y123\sqrt[3]{y^{12}}3y12​ to y4y^4y4.y123=y4 \sqrt[3]{y^{12}} = y^4 3y12​=y4

STEP 8

The expression now simplifies to:y4⋅y23 y^4 \cdot \sqrt[3]{y^2} y4⋅3y2​

STEP 9

This is the simplified form of the original expression by factoring. The final answer is:y143=y4⋅y23 \sqrt[3]{y^{14}} = y^4 \cdot \sqrt[3]{y^2} 3y14​=y4⋅3y2​

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Question Simplify $\sqrt[3]{y^{14}}$ by factoring, where $y$ is a positive real number and the radicand does not contain negative quantities raised to even powers.

Assumptions
1. The variable $y$ represents a positive real number.
2. The radicand does not contain negative quantities raised to even powers.
3. We are simplifying the expression $\sqrt[3]{y^{14}}$ by factoring.

Identify the index of the radical and the exponent of the variable within the radicand. The index of the radical is 3 (since it is a cube root), and the exponent of the variable $y$ is 14.

To simplify the expression by factoring, we look for the highest power of $y$ that is a multiple of the index of the radical. In this case, we are looking for the highest power of $y$ that is a multiple of 3.

Since $14 = 3 \times 4 + 2$ , the highest power of $y$ that is a multiple of 3 and less than or equal to 14 is $3 \times 4 = 12$ . We can write $y^{14}$ as $y^{12} \cdot y^2$ .

Now we can rewrite the original expression using the factorization from the previous step.
$\sqrt[3]{y^{14}} = \sqrt[3]{y^{12} \cdot y^2}$

We can now separate the radical into two parts using the property of radicals that states $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ .
$\sqrt[3]{y^{12} \cdot y^2} = \sqrt[3]{y^{12}} \cdot \sqrt[3]{y^2}$

Since $y^{12}$ is a perfect cube (because $12$ is a multiple of $3$ ), we can simplify $\sqrt[3]{y^{12}}$ to $y^4$ .
$\sqrt[3]{y^{12}} = y^4$

The expression now simplifies to:
$y^4 \cdot \sqrt[3]{y^2}$

This is the simplified form of the original expression by factoring. The final answer is:
$\sqrt[3]{y^{14}} = y^4 \cdot \sqrt[3]{y^2}$