Sure, let's go through the detailed step-by-step solution to simplify the expression
[tex]\[
\left(\frac{4 a^2 b^{-5}}{6 a^{-1} b^{-4}}\right)^3
\][/tex]
### Step 1: Simplify the Coefficients
We start by simplifying the coefficients, which are the constants of the numerator and the denominator:
[tex]\[
\frac{4}{6}
\][/tex]
This can be reduced by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
[tex]\[
\frac{4}{6} = \frac{2}{3}
\][/tex]
### Step 2: Simplify the Exponents for [tex]\( a \)[/tex]
Next, we handle the exponents of [tex]\( a \)[/tex]. In the numerator, the exponent of [tex]\( a \)[/tex] is 2, and in the denominator, the exponent of [tex]\( a \)[/tex] is -1. When dividing like bases, we subtract the exponents:
[tex]\[
a^{2 - (-1)} = a^{2 + 1} = a^3
\][/tex]
### Step 3: Simplify the Exponents for [tex]\( b \)[/tex]
Now, we simplify the exponents for [tex]\( b \)[/tex]. In the numerator, the exponent of [tex]\( b \)[/tex] is -5, and in the denominator, the exponent of [tex]\( b \)[/tex] is -4. When dividing like bases, we subtract the exponents:
[tex]\[
b^{-5 - (-4)} = b^{-5 + 4} = b^{-1}
\][/tex]
### Step 4: Combine the Simplified Expressions
We now put our simplified results together:
[tex]\[
\frac{2}{3} \cdot a^3 \cdot b^{-1}
\][/tex]
### Step 5: Raise to the Power of 3
Next, we need to raise the entire expression to the power of 3:
[tex]\[
\left(\frac{2}{3} \cdot a^3 \cdot b^{-1}\right)^3
\][/tex]
### Step 6: Apply Exponent to Each Component
We apply the exponent of 3 to each part of the expression separately:
#### Coefficient:
[tex]\[
\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}
\][/tex]
#### Exponent for [tex]\( a \)[/tex]:
[tex]\[
(a^3)^3 = a^{3 \cdot 3} = a^9
\][/tex]
#### Exponent for [tex]\( b \)[/tex]:
[tex]\[
(b^{-1})^3 = b^{-1 \cdot 3} = b^{-3}
\][/tex]
### Step 7: Combine Final Expression
Putting it all together, the simplified form is:
[tex]\[
\frac{8}{27} \cdot a^9 \cdot b^{-3}
\][/tex]
Or more concisely:
[tex]\[
\frac{8 a^9}{27 b^3}
\][/tex]
In decimal form, the constant coefficient is approximately [tex]\( 0.2962962962962962 \)[/tex]. Thus, expressed in a different format, the final simplified expression is:
[tex]\[
0.2962962962962962 \cdot a^9 \cdot b^{-3}
\][/tex]
And there you have the complete, detailed solution!
