To solve the expression \(\sqrt[8]{a^5 b} - \sqrt[7]{a^5 b} \cdot \sqrt[8]{a^5 b}\), we can follow these steps:
1. Convert Radicals to Exponents: Start by expressing the radicals using fractional exponents.
- \(\sqrt[8]{a^5 b}\) can be written as \((a^5 b)^{\frac{1}{8}}\).
- \(\sqrt[7]{a^5 b}\) can be written as \((a^5 b)^{\frac{1}{7}}\).
2. Rewrite the Expression:
- The given expression can be rewritten as \((a^5 b)^{\frac{1}{8}} - (a^5 b)^{\frac{1}{7}} \cdot (a^5 b)^{\frac{1}{8}}\).
3. Simplify the Product of Exponents: Using the property of exponents that \(x^m \cdot x^n = x^{m+n}\):
- \(\sqrt[7]{a^5 b} \cdot \sqrt[8]{a^5 b}\) becomes \((a^5 b)^{\frac{1}{7}} \cdot (a^5 b)^{\frac{1}{8}} = (a^5 b)^{\frac{1}{7} + \frac{1}{8}}\).
4. Sum the Exponents:
- Compute \(\frac{1}{7} + \frac{1}{8}\).
- The value \(\frac{1}{7} + \frac{1}{8}\) can be calculated as:
[tex]\[
\frac{1}{7} + \frac{1}{8} = \frac{8 + 7}{56} = \frac{15}{56}
\][/tex]
5. Rewrite the Expression:
- The expression thus simplifies to:
[tex]\[
(a^5 b)^{\frac{1}{8}} - (a^5 b)^{\frac{15}{56}}
\][/tex]
6. Convert the Exponents Back to Original Form:
- \((a^5 b)^{\frac{1}{8}} \) remains as it is, and \((a^5 b)^{\frac{15}{56}} \) simplifies further by converting it back to:
[tex]\[
(a^5 b)^{\frac{15}{56}} = (a^5 b)^{0.267857142857143}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
(a^5 b)^{0.125} - (a^5 b)^{0.267857142857143}
\][/tex]
This is the step-by-step solution to the problem.
