Simplify (32ab^4)/(-8a^-2b)-3a^3b^3
The question is asking for the simplification of a mathematical expression that involves algebraic fractions and exponents. The initial expression is a fraction $(32ab^4)/(-8a^-2b)$, which is followed by another term $-3a^3b^3$. The simplification process would likely involve steps such as:
Reducing the fraction by factoring out common terms or by using exponent rules.
Applying the negative exponent rule which states that a^-n = 1/a^n to the term $a^-2$in the denominator.
Multiplying or dividing terms with the same base by adding or subtracting their exponents, respectively.
Combining like terms if possible to reach the simplest form of the expression.
The question does not provide an equal sign followed by another expression or value, which indicates that it's not asking for the equation to be solved for a specific variable but merely for the manipulation and reduction of the expression given.
$\frac{32 a b^{4}}{- 8 a^{- 2} b} - 3 a^{3} b^{3}$
Answer
Solution:
Simplification Process
Step 1: Break down each component of the expression.
Step 1.1: Apply the rule for negative exponents, which states $a^{-n} = \frac{1}{a^n}$, to reposition $a^{-2}$ to the numerator.
$$ \frac{32ab^4 \cdot a^2}{-8b} - 3a^3b^3 $$
Step 1.2: Combine the exponents of $a$ by summing them up.
Step 1.2.1: Prepare to combine $a^2$ and $a$.
$$ \frac{32(a^2 \cdot a)b^4}{-8b} - 3a^3b^3 $$
Step 1.2.2: Execute the combination of $a^2$ and $a$.
Step 1.2.2.1: Express $a$ as $a^1$.
$$ \frac{32(a^2 \cdot a^1)b^4}{-8b} - 3a^3b^3 $$
Step 1.2.2.2: Utilize the exponent rule $a^m \cdot a^n = a^{m+n}$.
$$ \frac{32a^{2+1}b^4}{-8b} - 3a^3b^3 $$
Step 1.2.3: Add the exponents 2 and 1.
$$ \frac{32a^3b^4}{-8b} - 3a^3b^3 $$
Step 1.3: Simplify by canceling out common factors between 32 and -8.
Step 1.3.1: Extract the factor of 8 from $32a^3b^4$.
$$ \frac{8(4a^3b^4)}{-8b} - 3a^3b^3 $$
Step 1.3.2: Eliminate the shared factors.
Step 1.3.2.1: Factor out 8 from $-8b$.
$$ \frac{8(4a^3b^4)}{8(-b)} - 3a^3b^3 $$
Step 1.3.2.2: Cancel the common factor of 8.
$$ \frac{\cancel{8}(4a^3b^4)}{\cancel{8}(-b)} - 3a^3b^3 $$
Step 1.3.2.3: Rewrite the simplified expression.
$$ \frac{4a^3b^4}{-b} - 3a^3b^3 $$
Step 1.4: Reduce the expression by canceling out common factors of $b^4$ and $b$.
Step 1.4.1: Factor out $b$ from $4a^3b^4$.
$$ \frac{b(4a^3b^3)}{-b} - 3a^3b^3 $$
Step 1.4.2: Distribute the negative sign across the fraction.
$$ -1 \cdot (4a^3b^3) - 3a^3b^3 $$
Step 1.5: Express $-1 \cdot (4a^3b^3)$ as $-(4a^3b^3)$.
$$ -(4a^3b^3) - 3a^3b^3 $$
Step 1.6: Multiply 4 by $-1$.
$$ -4a^3b^3 - 3a^3b^3 $$
Step 2: Combine like terms.
$$ -7a^3b^3 $$
Knowledge Notes:
To simplify the given expression, several algebraic rules and properties are applied:
Negative Exponent Rule: For any non-zero number $a$ and integer $n$, $a^{-n} = \frac{1}{a^n}$. This rule allows us to move factors with negative exponents from the denominator to the numerator or vice versa by changing the sign of the exponent.
Combining Like Terms: Terms that have identical variable parts can be combined by adding or subtracting their coefficients.
Multiplication of Powers with the Same Base: When multiplying powers with the same base, we add the exponents, as per the rule $a^m \cdot a^n = a^{m+n}$.
Division of Powers with the Same Base: When dividing powers with the same base, we subtract the exponents, as per the rule $a^m / a^n = a^{m-n}$.
Simplifying Fractions: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out.
Distributive Property: This property is used when multiplying a number by a sum or difference, as in $a(b \pm c) = ab \pm ac$.
By applying these rules systematically, we can simplify the given algebraic expression to its simplest form.
