91Ó°ÊÓ

The power rule is a fundamental idea in algebra involving exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. Mathematically, this is expressed as \((a^m)^n = a^{m \times n}\).
Let's break down our exercise using the power rule.
Consider the expression \((b^{7})^{5}\). Applying the power rule, we multiply 7 by 5, giving us \((b^{35})\).
Similarly, for \((b^{2})^{6}\), you multiply 2 by 6, resulting in \((b^{12})\).
Remember: the power rule helps us condense multiple layers of exponents into a single, more manageable exponent.

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. For example, \(b^2 \) means \(b \times b\).
In our problem, we were given \(b^7\) and \(b^2\) within the parentheses. Exponents follow specific rules that make complex problems simpler.
One of these rules, as we've discussed, is the power rule. There are others, including:

• \(a^m \times a^n = a^{m+n} \): When multiplying numbers with the same base, add their exponents.
• \(a^0 = 1 \): Any base raised to the power of zero is 1.
• \(a^{-m} = \frac{1}{a^m} \): Negative exponents represent the reciprocal of the base with a positive exponent.

Knowing these rules can make handling exponents much easier, allowing you to tackle complex algebraic expressions with confidence.

Simplification is the process of making an expression easier to understand or work with. In algebra, it usually involves combining like terms, reducing fractions, and applying mathematical rules.
In our exercise, we started with the expression \( (b^{7})^{5} \times (b^{2})^{6} \) and aimed to simplify it. By applying the power rule, we first obtained \( b^{35} \) and \( b^{12} \).
Next, we used the rule for multiplying exponents with the same base, which allowed us to combine the exponents:

• \ b^{35 + 12} = b^{47} \

The final result, \ b^{47}\, is much simpler than what we started with. This process of simplifying makes it easier to evaluate, interpret, and work with algebraic expressions in general.
Using these techniques not only helps in solving homework problems but also builds a strong foundation for more advanced mathematical concepts.