Problem 8 - Elementary Algebra

Question

Make use of either or both the power rule for products and the power rule for powers to simplify each expression.] $$ [(a+8)(a+5)]^{4} $$

Step-by-Step Solution

Answer

Question: Simplify and express the given expression in its simplest form: \([(a+8)(a+5)]^{4}\) Answer: \((a^8 + 28561a^4 + 2560000)\)

Step 1: Expand the product of the two binomials

First, we need to expand the expression \((a+8)(a+5)\). To do this, we will use the distributive property (also known as the FOIL method): \((a+8)(a+5) = a(a+5) + 8(a+5)\) Now let's distribute the a and 8 to both terms inside the parentheses: \(a(a+5) = a^2 + 5a\) \(8(a+5) = 8a + 40\) Now, combine the results: \((a+8)(a+5) = a^2 + 5a + 8a + 40 = a^2 + 13a + 40\) At this point, we have expanded the product of the two binomials.

Step 2: Apply the power rule for powers

Now that we have simplified the expression within the brackets, we need to raise the result to the power of 4: \([(a^2 + 13a + 40)]^{4}\) To do this, we will need to apply the power rule for powers. This means that each term in the expression will be raised to the power of 4: \((a^2)^4 = a^8\) \((13a)^4 = 28561a^4\) \((40)^4 = 2560000\) Now, we will reassemble the expression with the new powers: \([(a+8)(a+5)]^{4} = (a^8 + 28561a^4 + 2560000)\) The simplified expression is: $$ (a^8 + 28561a^4 + 2560000) $$

Key Concepts

Power Rule for Products

When you are dealing with multiplying expressions with exponents, the power rule for products can be your best ally. This nifty rule states that when multiplying two powers that have the same base, you simply add the exponents together. For instance, if you have \(x^m * x^n\), the result will be \(x^{m+n}\).

Let's say you have \((3x^2)(4x^3)\). You multiply the constants 3 and 4 to get 12, and for the variable x, you simply add the exponents: 2 + 3 equals 5, resulting in \(12x^5\). Always remember that this rule applies only when the base remains the same; if the bases are different, this rule is not applicable.

Power Rule for Powers

When you're working with an exponent raised to another exponent, it's time to invoke the power rule for powers. This rule simplifies life by saying that you multiply the exponents together. So, if you have an expression like \((x^a)^b\), you would wind up with \(x^{a*b}\).

For example, consider \((2^3)^2\). You would multiply the exponents 3 and 2 to get 6, and the expression would simplify to \(2^6\). This is particularly useful when dealing with algebraic expressions where variables have exponentiation, like \((a^2)^4\) turning into \(a^8\), by multiplying 2 and 4.

Distributive Property

The distributive property is a cornerstone of algebra and allows us to simplify and expand expressions. It tells us that you can 'distribute' a multiplier over terms within parentheses. Mathematically, it looks like this: \(a(b + c) = ab + ac\). Whenever you see an expression like \(2(x + 3)\), you apply the distributive property to get \(2*x + 2*3\) which simplifies to \(2x + 6\).

This is especially helpful when expanding binomials or complex polynomials. For example, with \((x + 1)(x + 2)\), you distribute each term of the first binomial through the second, resulting in \(x(x + 2) + 1(x + 2)\), which ultimately simplifies to \(x^2 + 3x + 2\).

FOIL Method

The FOIL method is a specific case of the distributive property that's used for multiplying two binomials together. FOIL stands for First, Outer, Inner, Last, and it refers to the order in which you multiply the terms in the binomials.

First:

Multiply the first terms in each binomial.

Outer:

Multiply the outermost terms in the product.

Inner:

Multiply the inner terms.

Last:

Multiply the last terms.

For example, multiply \((a + b)(c + d)\):

• First: \(a * c\)
• Outer: \(a * d\)
• Inner: \(b * c\)
• Last: \(b * d\)

Then, you add all these products together to get \(ac + ad + bc + bd\). FOIL is a quick and easy way to expand binomials without missing any terms.