(A^-7 b^8/9)/a^9 b^-2/3
To simplify the expression \(\frac{{(A^{-7} \cdot b^{8/9})}}{{a^9 \cdot b^{-2/3}}}\), we can use the properties of exponents.
First, let's simplify the numerator:
\((A^{-7} \cdot b^{8/9})\)
Using the property \(a^{-n} = \frac{1}{{a^n}}\), we can rewrite \(A^{-7}\) as \(\frac{1}{{A^7}}\). The expression now becomes
\(\frac{{\frac{1}{{A^7}} \cdot b^{8/9}}}{{a^9 \cdot b^{-2/3}}}\)
Next, let's simplify the denominator:
\(a^9 \cdot b^{-2/3}\)
Using the property \(\frac{a^m}{a^n} = a^{m-n}\), we can rewrite \(b^{-2/3}\) as \(\frac{1}{{b^{2/3}}}\). The expression now becomes
\(\frac{{\frac{1}{{A^7}} \cdot b^{8/9}}}{{a^9 \cdot \frac{1}{{b^{2/3}}}}}\)
Now, let's simplify the expression further by multiplying the numerators and denominators:
\(\frac{{1 \cdot b^{8/9}}}{{A^7 \cdot a^9 \cdot \frac{1}{{b^{2/3}}}}}\)
Using the property \(\frac{a^m}{a^n} = a^{m-n}\) again, we can rewrite \(b^{8/9}\) as \(b^{8/9 - 2/3}\):
\(\frac{{b^{8/9 - 2/3}}}{{A^7 \cdot a^9 \cdot \frac{1}{{b^{2/3}}}}}\)
To simplify the exponents, we need a common denominator:
\(\frac{{b^{24/27 - 18/27}}}{{A^7 \cdot a^9 \cdot \frac{1}{{b^{2/3}}}}}\)
\(\frac{{b^{6/27}}}{{A^7 \cdot a^9 \cdot \frac{1}{{b^{2/3}}}}}\)
Further simplifying the expression:
\(\frac{{b^{2/9}}}{{A^7 \cdot a^9 \cdot \frac{1}{{b^{2/3}}}}}\)
Using the property \(\frac{1}{a} = a^{-1}\), we can rewrite \(\frac{1}{{b^{2/3}}}\) as \(b^{-2/3}\):
\(\frac{{b^{2/9}}}{{A^7 \cdot a^9 \cdot b^{-2/3}}}\)
Using the property \(a^m \cdot a^n = a^{m+n}\), we can combine the \(b\) terms:
\(\frac{{b^{2/9 - -2/3}}}{{A^7 \cdot a^9}}\)
Simplifying the exponents:
\(\frac{{b^{6/9 + 6/9}}}{{A^7 \cdot a^9}}\)
\(\frac{{b^{12/9}}}{{A^7 \cdot a^9}}\)
Now, let's simplify the exponent:
\(b^{12/9} = b^{4/3}\)
Finally, the expression is simplified to:
\(\frac{{b^{4/3}}}{{A^7 \cdot a^9}}\)
To simplify the expression ((A^(-7) b^(8/9)) / (a^9 b^(-2/3)), you can follow these steps:
Step 1: Combine the exponents of like bases.
In this case, the bases are A and a, and the exponent of A is -7, while the exponent of a is 9. So, we subtract the exponents: A^(-7) / a^9.
Similarly, for the base b, the exponent is (8/9) - (-2/3), which can be simplified as 8/9 + 2/3. Find a common denominator, which in this case is 9. So, (8/9) + (2/3) = (8/9) + (6/9) = 14/9. The expression now becomes A^(-7) / a^9 * b^(14/9).
Step 2: Simplify the expression.
To simplify A^(-7) / a^9 * b^(14/9), we can rewrite A^(-7) as 1/A^7 and a^9 as 1/a^9.
The expression now becomes (1/A^7) / (1/a^9) * b^(14/9).
Step 3: Invert and multiply the fractions.
Dividing by a fraction is equivalent to multiplying by the reciprocal of that fraction. So, we can rewrite the expression as (1/A^7) * (a^9/1) * b^(14/9).
Step 4: Combine the terms.
Multiply the variables with the same base by adding the exponents. In this case, A and a have different bases, so they cannot be combined.
The expression simplifies to (a^9/A^7) * b^(14/9).
Step 5: Rearrange the terms.
To express the result in a simpler form, we can rearrange the terms.
The simplified expression is b^(14/9) * a^9/A^7.
Therefore, ((A^(-7) b^(8/9)) / (a^9 b^(-2/3)) simplifies to b^(14/9) * a^9/A^7.