Simplify Algebraic Expression with Exponents

The problem requires us to simplify the following mathematical expression involving multiplication and exponents:

$(a^{-10} \times b^{10} \times c^{-9}) \times (a^0 \times b^7 \times c^6)$

Understanding Exponent Rules for Multiplication

To simplify this expression, we rely on fundamental rules of exponents. The key rule here is the product rule, which states that when multiplying two exponential terms with the same base, we add their exponents:

$x^m \times x^n = x^{m+n}$

Additionally, we use the rule that any non-zero base raised to the power of zero equals 1:

$x^0 = 1$ (where $x \neq 0$)

Step-by-Step Simplification Process

Let's simplify the given expression by applying these rules systematically:

1. Group terms with the same base: Rearrange the expression to gather all terms with base $a$ together, all terms with base $b$ together, and all terms with base $c$ together.

   The expression becomes:

   $(a^{-10} \times a^0) \times (b^{10} \times b^7) \times (c^{-9} \times c^6)$

2. Apply the product rule for exponents: Now, apply the rule $x^m \times x^n = x^{m+n}$ to each group.

   • For the base $a$: The exponents are $-10$ and $0$. Adding them gives $a^{-10 + 0} = a^{-10}$.
   • For the base $b$: The exponents are $10$ and $7$. Adding them gives $b^{10 + 7} = b^{17}$.
   • For the base $c$: The exponents are $-9$ and $6$. Adding them gives $c^{-9 + 6} = c^{-3}$.

3. Combine the simplified bases: Put the results for each base back together to form the final simplified expression.

   The simplified expression is: $a^{-10} \times b^{17} \times c^{-3}$

Final Simplified Expression

After applying the rules of exponents, the simplified form of the expression is:

$a^{-10} \times b^{17} \times c^{-3}$