Simplify Exponential Expressions

Watch and learn how to simplify an exponential expression with negative exponents.

Simplifying $$a^4 \times a^{-2} \times b^{-6} \times b^3$$

Simplifying expressions with exponents involves applying fundamental rules of exponentiation to condense the terms into a more manageable form. Let's break down the steps to simplify the expression $$a^4 \times a^{-2} \times b^{-6} \times b^3$$.

Step 1: Combine Like Bases

The initial step is to identify and combine the terms with similar bases. In this expression, we have bases $$a$$ and $$b$$.

Step 2: Apply the Laws of Exponents

With the like bases identified, we apply the laws of exponents to consolidate the terms. When multiplying powers with the same base, we add their exponents.

For the terms involving $$a$$, namely $$a^4 \times a^{-2}$$, we add their exponents: $$4 + (-2) = 2$$. Consequently, $$a^4 \times a^{-2}$$ simplifies to $$a^2$$.

Similarly, for the terms involving $$b$$, specifically $$b^{-6} \times b^3$$, we sum their exponents: $$(-6) + 3 = -3$$. Hence, $$b^{-6} \times b^3$$ simplifies to $$b^{-3}$$.

Step 3: Combine Simplified Terms

Having simplified each base, we recombine them into a single expression. Thus, the simplified expression becomes $$a^2 \times b^{-3}$$.

Step 4: Final Simplification

It's crucial to note that a negative exponent signifies the reciprocal of the base. Therefore, $$b^{-3}$$ is equivalent to $$\frac{1}{b^3}$$.

Consequently, the final simplified expression is $$a^2 \times \frac{1}{b^3}$$.

By following these systematic steps, we can efficiently simplify expressions involving exponents, aiding in clearer understanding and easier manipulation of mathematical equations.