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When working with algebraic expressions, understanding the laws of exponents is crucial. Exponents are a way to express repeated multiplication of the same number. For example, \( b^3 \) means \( b \times b \times b \).
In algebraic expressions like the one given, different powers of the same variable are often found together. To simplify these when multiplied, you can apply the rule: add the exponents.
For instance, if you are multiplying \( b^3 \), \( b^2 \), and \( b^4 \), you will sum the exponents:

• Exponent of \( b \) in \( b^3 \) is 3.
• Exponent of \( b \) in \( b^2 \) is 2.
• Exponent of \( b \) in \( b^4 \) is 4.

Thus, the combined exponent is \( 3 + 2 + 4 = 9 \), resulting in \( b^9 \). Remember, this rule only applies to terms with the same base.

Coefficients are the numerical factors that multiply the variables in an algebraic expression. Understanding how to multiply these numbers correctly is vital.
Consider the example where you have

• \(-4\)
• \(\frac{1}{6}\)
• \(-9\)

To solve, first multiply the negative numbers: \(-4 \times -9\). Since multiplying two negative numbers gives a positive result, this equals 36. Finally, multiply the result by \(\frac{1}{6}\):
\[36 \times \frac{1}{6} = 6\]
The final answer for the coefficients is 6. It's like multiplying numbers without variables; just follow arithmetic rules carefully.

Polynomial expressions are algebraic expressions that can have constants, variables, and exponents. They might not look too intimidating, but they can be complex.
This particular expression has multiple terms, each with a coefficient and a power of the base 'b'. When simplifying:

• Concentrate first on multiplying all numerical coefficients together.
• Then, use the laws of exponents to handle the variable parts.

A polynomial becomes simpler as you combine its like terms (if adding or subtracting) or use multiplication rules to merge terms by combining exponents and coefficients like in the given expression.
Simplifying helps reduce expressions to their most compact form, making them easier to work with. In this exercise, we simplify the polynomial \(-4 b^3 \times \frac{1}{6} b^2 \times -9 b^4\) to \(6b^9\). Thus, we find a neater version that's straightforward to manage.