At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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!
[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!
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
