To simplify the expression [tex]\(\frac{\sqrt{8}}{1 - \sqrt{3z}}\)[/tex] by rationalizing the denominator, let's follow these steps:
1. Write down the original expression:
[tex]\[
\frac{\sqrt{8}}{1 - \sqrt{3z}}
\][/tex]
2. Find the conjugate of the denominator:
The conjugate of [tex]\(1 - \sqrt{3z}\)[/tex] is [tex]\(1 + \sqrt{3z}\)[/tex].
3. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{\sqrt{8}}{1 - \sqrt{3z}} \cdot \frac{1 + \sqrt{3z}}{1 + \sqrt{3z}}
\][/tex]
4. Use the difference of squares to simplify the denominator:
The denominator becomes:
[tex]\[
(1 - \sqrt{3z})(1 + \sqrt{3z}) = 1^2 - (\sqrt{3z})^2 = 1 - 3z
\][/tex]
So the expression is now:
[tex]\[
\frac{\sqrt{8} (1 + \sqrt{3z})}{1 - 3z}
\][/tex]
5. Distribute [tex]\(\sqrt{8}\)[/tex] in the numerator:
[tex]\[
\sqrt{8} (1 + \sqrt{3z}) = \sqrt{8} + \sqrt{8} \cdot \sqrt{3z} = \sqrt{8} + \sqrt{24z}
\][/tex]
Hence, the simplified expression is:
[tex]\[
\frac{\sqrt{8} + \sqrt{24z}}{1 - 3z}
\][/tex]
6. Identify coefficients and constants in the simplified form:
- [tex]\(a\)[/tex] is the coefficient of the square roots in the numerator, which is [tex]\(2\)[/tex], since [tex]\(\sqrt{8} = 2\sqrt{2}\)[/tex] and [tex]\(\sqrt{24z} = 2\sqrt{6z}\)[/tex].
- [tex]\(b\)[/tex] is the radicand without variables in the square root term [tex]\(\sqrt{24z}\)[/tex], which simplifies to [tex]\(6\)[/tex].
- [tex]\(d\)[/tex] is the radicand without variables in [tex]\(\sqrt{8}\)[/tex], which simplifies to [tex]\(2\)[/tex].
- [tex]\(e\)[/tex] is [tex]\(1\)[/tex], the constant term in the denominator.
- [tex]\(f\)[/tex] is the coefficient of [tex]\(z\)[/tex] in the denominator term [tex]\(-3z\)[/tex], which is [tex]\(-3\)[/tex].
So, the final values are:
- Our value for [tex]\(a\)[/tex] is [tex]\(2\)[/tex]
- Our value for [tex]\(b\)[/tex] is [tex]\(6\)[/tex]
- Our value for [tex]\(d\)[/tex] is [tex]\(2\)[/tex]
- Our value for [tex]\(e\)[/tex] is [tex]\(1\)[/tex]
- Our value for [tex]\(f\)[/tex] is [tex]\(-3\)[/tex]
