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Answer :
To simplify the expression [tex]\(\frac{\sqrt{49} \cdot y^{1 / 2}}{x^{-6}}\)[/tex], follow these steps:
1. Simplify [tex]\(\sqrt{49}\)[/tex]:
[tex]\[
\sqrt{49} = 7
\][/tex]
The square root of 49 is 7.
2. Simplify [tex]\(y^{1/2}\)[/tex]:
[tex]\[
y^{1/2} = \sqrt{y}
\][/tex]
When you raise a number to the power of [tex]\(\frac{1}{2}\)[/tex], it is the same as taking the square root of that number.
3. Simplify [tex]\(x^{-6}\)[/tex]:
[tex]\[
x^{-6} = \frac{1}{x^6}
\][/tex]
A negative exponent indicates a reciprocal, so [tex]\(x^{-6}\)[/tex] becomes [tex]\(\frac{1}{x^6}\)[/tex].
4. Rewrite the original expression:
Put everything together in the expression:
[tex]\[
\frac{\sqrt{49} \cdot y^{1/2}}{x^{-6}} = \frac{7 \cdot \sqrt{y}}{\frac{1}{x^6}}
\][/tex]
5. Simplify the division by a fraction:
Dividing by a fraction is the same as multiplying by its reciprocal, so:
[tex]\[
\frac{7 \cdot \sqrt{y}}{\frac{1}{x^6}} = 7 \cdot \sqrt{y} \cdot x^6
\][/tex]
Which simplifies to:
[tex]\[
7x^6\sqrt{y}
\][/tex]
Therefore, the simplified expression is [tex]\(7x^6\sqrt{y}\)[/tex].
The correct answer is option A. [tex]\(7 x^6 \sqrt{y}\)[/tex].
1. Simplify [tex]\(\sqrt{49}\)[/tex]:
[tex]\[
\sqrt{49} = 7
\][/tex]
The square root of 49 is 7.
2. Simplify [tex]\(y^{1/2}\)[/tex]:
[tex]\[
y^{1/2} = \sqrt{y}
\][/tex]
When you raise a number to the power of [tex]\(\frac{1}{2}\)[/tex], it is the same as taking the square root of that number.
3. Simplify [tex]\(x^{-6}\)[/tex]:
[tex]\[
x^{-6} = \frac{1}{x^6}
\][/tex]
A negative exponent indicates a reciprocal, so [tex]\(x^{-6}\)[/tex] becomes [tex]\(\frac{1}{x^6}\)[/tex].
4. Rewrite the original expression:
Put everything together in the expression:
[tex]\[
\frac{\sqrt{49} \cdot y^{1/2}}{x^{-6}} = \frac{7 \cdot \sqrt{y}}{\frac{1}{x^6}}
\][/tex]
5. Simplify the division by a fraction:
Dividing by a fraction is the same as multiplying by its reciprocal, so:
[tex]\[
\frac{7 \cdot \sqrt{y}}{\frac{1}{x^6}} = 7 \cdot \sqrt{y} \cdot x^6
\][/tex]
Which simplifies to:
[tex]\[
7x^6\sqrt{y}
\][/tex]
Therefore, the simplified expression is [tex]\(7x^6\sqrt{y}\)[/tex].
The correct answer is option A. [tex]\(7 x^6 \sqrt{y}\)[/tex].
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