Problem 103 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ 6 a^{3} b^{3}\left(4 a^{2} b^{6}+7 a b^{8}+2 b^{10}+14\right) $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(6a^{3}b^{3}(4a^{2}b^{6}+7ab^{8}+2b^{10}+14)\) Answer: \(24a^{5}b^{9} + 42a^{4}b^{11} + 12a^{3}b^{13} + 84a^{3}b^{3}\)

Step 1: Distribute the term across the polynomial

Multiply the term \(6a^{3}b^{3}\) by each term inside the parentheses: $$ 6a^{3}b^{3}(4a^{2}b^{6}) + 6a^{3}b^{3}(7ab^{8}) + 6a^{3}b^{3}(2b^{10}) + 6a^{3}b^{3}(14) $$

Step 2: Apply the product of powers property

Use the product of powers property to simplify each term. The property states that \(a^{m}a^{n}=a^{m+n}\) and \(b^{m}b^{n}=b^{m+n}\). Apply this to each product we got in step 1: $$ 24a^{5}b^{9} + 42a^{4}b^{11} + 12a^{3}b^{13} + 84a^{3}b^{3} $$

Step 3: Combine like terms

In the resulting expression, there are no like terms to combine (no terms with the same variables and exponents). Thus, this is the simplified and final expression: $$ 24a^{5}b^{9} + 42a^{4}b^{11} + 12a^{3}b^{13} + 84a^{3}b^{3} $$

Key Concepts

Product of Powers Property

When multiplying powers with the same base, you add their exponents. This simple rule is known as the product of powers property and is expressed as \(a^m \times a^n = a^{m+n}\). Imagine you have \(a^3\) and you want to multiply it by \(a^4\). According to the product of powers property, the answer would be \(a^{3+4} = a^7\). This rule simplifies multiplication when dealing with variables raised to powers. In our example, this property helps us reduce expressions like \(a^3 \times a^2\) to \(a^5\). Here, both bases are the same \(a\), so their exponents are simply added together. Similarly, for \(b^3 \times b^6\), the result would be \(b^{3+6} = b^9\). Remember to apply this property to each term when distributing a monomial across a polynomial.

Distributive Property

The distributive property allows you to multiply a single term by each term inside a set of parentheses. It can be represented by the equation \(a(b+c) = ab + ac\). This becomes especially useful when dealing with polynomials. Imagine distributing a term like \(a\) over \(b+c+d\). You would multiply \(a\) separately by \(b\), \(c\), and \(d\), resulting in \(ab + ac + ad\). This process is often the first step in solving polynomial multiplication problems. In our given problem, \(6a^3b^3\) multiplies each term inside the parentheses: \(4a^2b^6\), \(7ab^8\), \(2b^{10}\), and \(14\). The distribution results in separate products that can be further simplified using other properties.

Combining Like Terms

In polynomial expressions, 'like terms' are terms that have identical variable parts, though their coefficients (numerical parts) may differ. For example, \(3x^2\) and \(5x^2\) are like terms because their variables and exponents are the same: \(x^2\). Combining like terms means adding or subtracting their coefficients. You do not change the variables or their exponents. For instance, \(3x^2 + 5x^2\) simplifies to \(8x^2\).

• Identify terms with the same variables and exponents.
• Add or subtract their coefficients.
• Retain the variable and its exponent.

In our exercise, after multiplying and simplifying, check for like terms in the result. In this case, each term in the result is distinct, as there are no terms with both the same variables and exponents, so nothing needs to be combined.