Simplify a^-3b^2(ba^3+b^-1a^2+b^-2a)

The problem is asking to simplify a mathematical expression that consists of variables (a and b) raised to various powers. To do this, you'll need to apply the laws of exponents which include rules such as multiplying powers with the same base by adding the exponents, dividing powers by subtracting the exponents, and raising a power to a power by multiplying the exponents. The expression also includes terms inside parentheses that are being multiplied or added to each other, which means you'll likely need to use the distributive property to expand these terms before simplifying. The goal is to combine like terms and reduce the expression to its simplest form.

$a^{- 3} b^{2} (b a^{3} + b^{- 1} a^{2} + b^{- 2} a)$

Answer

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Solution:

Step:1

Apply the negative exponent rule $a^{- n} = \frac{1}{a^{n}}$ to the initial expression.

$a^{- 3} b^{2} (b a^{3} + b^{- 1} a^{2} + b^{- 2} a) = \frac{b^{2}}{a^{3}} (b a^{3} + \frac{1}{b} a^{2} + \frac{1}{b^{2}} a)$

Step:2

Begin simplification of the terms.

Step:2.1

Combine the fraction $\frac{b^{2}}{a^{3}}$ with each term inside the parentheses.

$\frac{b^{2}}{a^{3}} (b a^{3} + \frac{1}{b} a^{2} + \frac{1}{b^{2}} a)$

Step:2.2

Simplify each term within the parentheses.

Step:2.2.1

Multiply $b^{2}$ by $b a^{3}$ .

$\frac{b^{2}}{a^{3}} (b a^{3}) = b^{3}$

Step:2.2.2

Multiply $b^{2}$ by $\frac{1}{b} a^{2}$ .

$\frac{b^{2}}{a^{3}} (\frac{1}{b} a^{2}) = b a^{2}$

Step:2.2.3

Multiply $b^{2}$ by $\frac{1}{b^{2}} a$ .

$\frac{b^{2}}{a^{3}} (\frac{1}{b^{2}} a) = a$

Step:3

Combine the simplified terms.

$\frac{b^{3} + b a^{2} + a}{a^{3}}$

Step:4

Reduce the expression by cancelling out common factors.

Step:4.1

Factor out $a$ from the denominator $a^{3}$ .

$\frac{b^{3} + b a^{2} + a}{a \cdot a^{2}}$

Step:4.2

Cancel out the common $a$ factor from the numerator and the denominator.

$\frac{b^{3} + b a^{2} + a}{a^{2}}$

Step:4.3

The final simplified expression is:

$\frac{b^{3}}{a^{2}} + b + \frac{1}{a}$

Knowledge Notes:

Negative Exponent Rule: $a^{- n} = \frac{1}{a^{n}}$ , which means that a negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent.
Simplification of Terms: In algebra, simplification involves combining like terms, factoring, expanding, and reducing expressions to their simplest form.
Distributive Property: $a (b + c) = a b + a c$ , which allows you to multiply a single term by each term within a parenthesis.
Combining Like Terms: Terms with the same variables raised to the same power can be combined by adding or subtracting their coefficients.
Cancelling Common Factors: When a factor appears in both the numerator and the denominator of a fraction, it can be cancelled out, simplifying the expression.

In this problem, we applied these principles to simplify an algebraic expression involving negative exponents and variable terms. The goal was to rewrite the expression in its simplest form by performing operations step by step, ensuring that each action adhered to the rules of algebra.