To simplify the expression [tex]\(\sqrt{48 y^{14}}\)[/tex], let's proceed step-by-step:
1. Express the radicand as a product of perfect squares and other factors:
Recognize that [tex]\(48\)[/tex] can be factored into [tex]\(16 \times 3\)[/tex], where [tex]\(16\)[/tex] is a perfect square:
[tex]\[
48 = 16 \times 3
\][/tex]
So the expression becomes:
[tex]\[
\sqrt{48 y^{14}} = \sqrt{16 \times 3 \times y^{14}}
\][/tex]
2. Separate the square roots of each factor:
Use the property of square roots [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex]:
[tex]\[
\sqrt{16 \times 3 \times y^{14}} = \sqrt{16} \times \sqrt{3} \times \sqrt{y^{14}}
\][/tex]
3. Take the square root of the perfect square numbers and even powers:
- The square root of [tex]\(16\)[/tex] is [tex]\(4\)[/tex]:
[tex]\[
\sqrt{16} = 4
\][/tex]
- For [tex]\(y^{14}\)[/tex], since [tex]\(14\)[/tex] is an even number, the square root simply halves the exponent:
[tex]\[
\sqrt{y^{14}} = y^{14/2} = y^7
\][/tex]
4. Combine all the simplified components:
Now, put all the simplified parts together:
[tex]\[
\sqrt{16} \times \sqrt{3} \times \sqrt{y^{14}} = 4 \times \sqrt{3} \times y^7
\][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{48 y^{14}}\)[/tex] is:
[tex]\[
4 \sqrt{3} y^7
\][/tex]
This completes the simplification process.
