Problem 43 - Algebra 1

Question

Evaluate the expression for the given values of the variables. $$ (a-b)^{4} \text { when } a=4 \text { and } b=2 $$

Step-by-Step Solution

Answer

The result of the expression \((a-b)^4\) when \(a=4\) and \(b=2\) is 16.

Step 1: Understand the problem

The exercise is asking to evaluate the expression \((a-b)^4\) with given values \(a=4\) and \(b=2\).

Step 2: Expand the Binomial

Use binomial theorem to expand the expression. The binomial theorem states \((a-b)^n = a^n - na^{n-1}b + \binom{n}{2}a^{n-2}b^2 - \binom{n}{3}a^{n-3}b^3 + \ldots + (-1)^nb^n\). Using this theorem, \((a-b)^4\) expands to \(a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4\).

Step 3: Substitute the given values

Now substitute the given values into the expanded equation, i.e. replace \(a\) with 4 and \(b\) with 2. This results in \(4^4 - 4*4^3*2 + 6*4^2*2^2 - 4*4*2^3 + 2^4\).

Step 4: Compute the solution

After substitution, simplify the arithmetic expression to find the answer. After simplifying the expression, the result is 16.

Key Concepts

Understanding Binomial Expansion

When faced with expressions like \( (a-b)^n \), the binomial expansion allows us to expand this expression into a polynomial. This can be very helpful for calculating powers without a calculator, especially when the power \( n \) is a small positive integer.

The binomial theorem gives us a formula:

• \( (a-b)^{n} = a^n - na^{n-1}b + \binom{n}{2}a^{n-2}b^2 - \ldots + (-1)^{n}b^n \)

This formula is made up of terms involving combinations, represented by \( \binom{n}{k} \). Each of these terms has specific coefficients determined by the combination numbers, distribution of powers, and the alternating sign.

Understanding binomial expansion is crucial in simplifying expressions to evaluate them efficiently, as we see in expanded polynomial form in our problem.

Meaning of Exponentiation

Exponentiation is a mathematical operation that involves raising a number to a power. This is shown as \( a^n \) where \( a \) is the base, and \( n \) is the exponent. In the expression \( (a-b)^4 \), \((a-b)\) is a base raised to the power of 4, or in other words, multiplied by itself 4 times.

To understand it simply, if you have \( 2^3 \), you multiply 2 by itself 3 times, equaling 8. In our specific scenario,

• \( (a-b)^4 = (a-b)(a-b)(a-b)(a-b) \)

Exponentiation plays a key role in the binomial expansion, as it dictates how we determine each component’s power within the expanded formula. This allows us to systematically break down higher powers into simpler calculations.

The Role of Substitution

In algebra, substitution is a technique used to replace variables with their given numerical values. This makes solving equations or evaluating expressions more manageable. For our exercise, we replace \( a \) with 4 and \( b \) with 2 in the expanded polynomial \( (a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4 \).

Substitution involves careful placement of values:

• \( 4^4 - 4 \times 4^3 \times 2 + 6 \times 4^2 \times 2^2 - 4 \times 4 \times 2^3 + 2^4 \)

It's key to replace variable symbols with the appropriate numbers before carrying out arithmetic operations. This step sets the stage for you to simplify and solve the expression more straightforwardly.

Evaluating Expressions

Once the expression is fully substituted with numerical values, the next step is to evaluate, or solve, the expression. Evaluation involves performing arithmetic operations (multiplication, addition, subtraction) to simplify the expression into a single number.

Following our substitution:

• Calculate each term: \( 4^4 = 256 \), \( 4 \times 4^3 \times 2 = 128 \times 2 = 256 \), \( 6 \times 4^2 \times 2^2 = 6 \times 16 \times 4 = 384 \) and so on.

Subtracting and adding these results gives the final answer:

• \( 256 - 256 + 384 - 128 + 16 = 272 \)

Understanding this allows solving complex expressions through systematic, simpler calculations, providing clear numerical answers efficiently.