91Ó°ÊÓ

When we talk about exponential form, we are referring to a mathematical way to represent numbers that involve repeated multiplication of the same number. This is shown as a base number raised to a power or an exponent. In our exercise, when we have the expression \(14^4\), it means "14 multiplied by itself 4 times." Here, 14 is the base, and 4 is the exponent. Writing an expression in exponential form is crucial because it simplifies complex multiplication into a more digestible notation. It allows us to appreciate the power of exponential functions, especially when dealing with larger computations, and provides a clear understanding of how many times the base number is used in the multiplication. For example:

• \(2^3\) means \(2 \cdot 2 \cdot 2\)
• \(5^2\) means \(5 \cdot 5\)

Binomial coefficients are the numerical factors that show up when a binomial expression is expanded. The binomial theorem helps us understand how to expand an expression like \((a+b)^n\). In practice, when expanding expressions such as \((10 + 4)^4\), binomial coefficients play a big role. These coefficients are calculated using the formula for combinations: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial, the product of an integer and all the integers below it. For the expression \((10+4)^4\), we determine the following binomial coefficients:

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

These coefficients help in expanding the binomial expression efficiently without manual repetitive addition. They denote how many ways we can select items from a group, crucial for understanding polynomial expansions.

Polynomial expansion involves breaking down a complex binomial expression into a sum or difference of simpler terms. Thanks to the binomial theorem, expanding expressions like \((a+b)^n\) becomes systematic and straightforward. In our case, expanding \((10 + 4)^4\) involves using the binomial coefficients to distribute and simplify each term.
Each term follows a specific pattern: \(\binom{n}{k} a^{n-k} b^k\), where you multiply the binomial coefficient by the appropriate powers of the two numbers. Here is how it is done:

• The first term is calculated as: \(1 \cdot 10^4 \cdot 4^0\)
• The second term is: \(4 \cdot 10^3 \cdot 4^1\)
• The third term is: \(6 \cdot 10^2 \cdot 4^2\)
• The fourth term is: \(4 \cdot 10^1 \cdot 4^3\)
• The fifth term is: \(1 \cdot 10^0 \cdot 4^4\)

Breaking these down helps simplify a complex expression into components that are easier to handle and compute, clearly showing the power of the binomial theorem for polynomial expansion.