 𝑎2/3.𝑏−5/7.(𝑎
 𝑏
 )7/8
 . 𝑎11/24
 𝑏23/56


We can simplify the expression \( \frac{a^{2/3}}{b^{5/7}} \cdot \left( \frac{a}{b} \right)^{7/8} \cdot a^{11/24} \cdot b^{23/56} \).

Let’s break it down step by step:

### Step 1: Simplify each component
- \( \frac{a^{2/3}}{b^{5/7}} \): This can be written as \( a^{2/3} \cdot b^{-5/7} \).
- \( \left( \frac{a}{b} \right)^{7/8} \): This is equivalent to \( a^{7/8} \cdot b^{-7/8} \).
- \( a^{11/24} \cdot b^{23/56} \) remains as is.

### Step 2: Combine powers of \( a \)
- Combine all the terms with \( a \):
  \[
  a^{2/3} \cdot a^{7/8} \cdot a^{11/24}
  \]
  Convert the fractions to a common denominator:
  \[
  \frac{2}{3} = \frac{16}{24}, \quad \frac{7}{8} = \frac{21}{24}, \quad \frac{11}{24} = \frac{11}{24}
  \]
  Now, add the exponents:
  \[
  \frac{16}{24} + \frac{21}{24} + \frac{11}{24} = \frac{48}{24} = 2
  \]
  So, the total power of \( a \) is \( a^2 \).

### Step 3: Combine powers of \( b \)
- Combine all the terms with \( b \):
  \[
  b^{-5/7} \cdot b^{-7/8} \cdot b^{23/56}
  \]
  Convert the fractions to a common denominator:
  \[
  \frac{-5}{7} = \frac{-40}{56}, \quad \frac{-7}{8} = \frac{-49}{56}, \quad \frac{23}{56} = \frac{23}{56}
  \]
  Now, add the exponents:
  \[
  \frac{-40}{56} + \frac{-49}{56} + \frac{23}{56} = \frac{-66}{56} = \frac{-33}{28}
  \]
  So, the total power of \( b \) is \( b^{-33/28} \).

### Final Expression:
\[
a^2 \cdot b^{-33/28}
\]

This is the simplified form of the expression.

---

Would you like more details on how any of these steps were performed, or do you have any questions?

### Related Questions:
1. How do you combine exponents with different bases?
2. What are the rules for simplifying expressions with fractional exponents?
3. How do you find a common denominator for fractions with different denominators?
4. Why does adding exponents work for multiplication of terms with the same base?
5. Can you apply similar rules to logarithms?

### Tip:
Always make sure to check the base when combining exponents; only terms with the same base can have their exponents added or subtracted.

𝑎^{2/3} / 𝑏^{−5/7} ⋅ (𝑎/𝑏)^{7/8} ⋅ 𝑎^{11/24} ⋅ 𝑏^{23/56}

This guide demonstrates how to simplify the expression a^{2/3} / b^{-5/7} ⋅ (a/b)^{7/8} ⋅ a^{11/24} ⋅ b^{23/56} using rules of exponents and algebra. Learn step-by-step to combine terms with fractional powers. Suitable for students in Grades 10-12.

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