Simplify (32ab^4)/(-8a^-2b)-3a^3b^3

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.

  $$\frac{32a{b}^4\cdot a^2}{-8b}-3a^3{b}^3$$

• Step 1.2: Combine the exponents of $a$ by summing them up.

  • Step 1.2.1: Prepare to combine $a^2$ and $a$.

    $$\frac{32(a^2\cdot a)b^4}{-8b}-3a^3{b}^3$$

  • Step 1.2.2: Execute the combination of $a^2$ and $a$.

    • Step 1.2.2.1: Express $a$ as $a^1$.

      $$\frac{32(a^2\cdot a^1)b^4}{-8b}-3a^3{b}^3$$

    • Step 1.2.2.2: Utilize the exponent rule $a^m\cdot a^n=a^{m+n}$.

      $$\frac{32a^{2+1}{b}^4}{-8b}-3a^3{b}^3$$

  • Step 1.2.3: Add the exponents 2 and 1.

    $$\frac{32a^3{b}^4}{-8b}-3a^3{b}^3$$

• Step 1.3: Simplify by canceling out common factors between 32 and -8.

  • Step 1.3.1: Extract the factor of 8 from $32a^3{b}^4$.

    $$\frac{8(4a^3{b}^4)}{-8b}-3a^3{b}^3$$

  • Step 1.3.2: Eliminate the shared factors.

    • Step 1.3.2.1: Factor out 8 from $-8b$.

      $$\frac{8(4a^3{b}^4)}{8(-b)}-3a^3{b}^3$$

    • Step 1.3.2.2: Cancel the common factor of 8.

      $$\frac{\cancel{8}(4a^3{b}^4)}{\cancel{8}(-b)}-3a^3{b}^3$$

    • Step 1.3.2.3: Rewrite the simplified expression.

      $$\frac{4a^3{b}^4}{-b}-3a^3{b}^3$$

• Step 1.4: Reduce the expression by canceling out common factors of $b^4$ and $b$.

  • Step 1.4.1: Factor out $b$ from $4a^3{b}^4$.

    $$\frac{b(4a^3{b}^3)}{-b}-3a^3{b}^3$$

  • Step 1.4.2: Distribute the negative sign across the fraction.

    $$-1\cdot (4a^3{b}^3)-3a^3{b}^3$$

• Step 1.5: Express $-1\cdot (4a^3{b}^3)$ as $-(4a^3{b}^3)$.

  $$-(4a^3{b}^3)-3a^3{b}^3$$

• Step 1.6: Multiply 4 by $-1$.

  $$-4a^3{b}^3-3a^3{b}^3$$

Step 2: Combine like terms.

$$-7a^3{b}^3$$

Knowledge Notes:

To simplify the given expression, several algebraic rules and properties are applied:

1. 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.

2. Combining Like Terms: Terms that have identical variable parts can be combined by adding or subtracting their coefficients.

3. Multiplication of Powers with the Same Base: When multiplying powers with the same base, we add the exponents, as per the rule $a^m\cdot a^n=a^{m+n}$.

4. Division of Powers with the Same Base: When dividing powers with the same base, we subtract the exponents, as per the rule $a^m/a^n=a^{m-n}$.

5. Simplifying Fractions: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out.

6. 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.