Problem 63 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ 5 m^{6}\left(2 m^{7}+3 m^{4}+m^{2}+m+1\right) $$

Step-by-Step Solution

Answer

Question: Simplify the following algebraic expression: \(5m^6(2m^7 + 3m^4 + m^2 + m + 1)\) Answer: The simplified form of the given algebraic expression is: \(10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6\)

Step 1: Identify the terms in parentheses

First, let's identify the terms inside the parentheses: $$ (2m^7 + 3m^4 + m^2 + m + 1) $$ These are the terms we will need to multiply by the monomial \(5m^6\).

Step 2: Apply the distributive property

Next, we will apply the distributive property by multiplying the monomial \(5m^6\) to each term inside the parentheses: $$ 5m^6 (2m^7) + 5m^6 (3m^4) + 5m^6 (m^2) + 5m^6 (m) + 5m^6 (1) $$

Step 3: Multiply the powers of m

Now, we will multiply the powers of m in each term. Remember, when multiplying monomials with the same base, we add their exponents: $$ (5 \cdot 2)m^{6+7} + (5 \cdot 3)m^{6+4} + 5m^{6+2} + 5m^{6+1} + 5m^6 $$

Step 4: Simplify the terms

Finally, we will simplify the terms in the expression: $$ 10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6 $$ The simplified form of the given algebraic expression is: $$ 10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6 $$

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra, where you distribute a single term over a sum or difference within parentheses. For instance, when you have an expression like \( a(b + c) \), the distributive property allows you to 'distribute' the \( a \) across the terms inside the parentheses, resulting in \( ab + ac \).

This property simplifies expressions and allows for further algebraic manipulation. It's especially useful when multiplying a monomial—like \( 5m^6 \)—by a polynomial—like \( (2m^7 + 3m^4 + m^2 + m + 1) \). By applying the distributive property, you multiply \( 5m^6 \) to each term within the parentheses. This technique plays a vital role in breaking down complex algebraic expressions into a more manageable form.

Multiplying Monomials

When multiplying monomials, two main rules come into play: multiply the coefficients (numerical parts) and then apply the laws of exponents for the variables. For instance, if you multiply \( 5m^6 \) by \( 2m^7 \), you would multiply the coefficients 5 and 2 to get 10. Then, to multiply the variables, you add the exponents, because the base (m) is the same in both terms, leading to \( m^{6+7} \) or \( m^{13} \).

Multiplying monomials is straightforward if you follow these steps consistently. Remember, your base, in this case \( m \), needs to be the same for the exponents to be added together; this is crucial when simplifying algebraic expressions involving powers.

Exponents

Exponents represent repeated multiplication, and understanding the rules governing exponents is crucial in algebra. For example, the expression \( m^6 \) indicates that the base \( m \) is multiplied by itself six times. When multiplying exponents with the same base, you simply add the exponents together. This is why \( m^6 \times m^7 \) becomes \( m^{6+7} \) which simplifies to \( m^{13} \).

Remember that the base must be the same, otherwise this rule does not apply. It's also important to note that an exponent of zero means the value is 1, because any number raised to the power of zero equals 1 (\( m^0 = 1 \)). Understanding these basics will help you manage expressions involving exponents more efficiently.

Algebraic Expression Simplification

Simplification of algebraic expressions involves reducing them to their simplest form without changing their value. The process includes several steps: identifying like terms, applying the distributive property, and then using laws of exponents when necessary. The goal is to make the expression as clear and concise as possible, making it easier to understand or to use in further calculations.

For the given algebraic expression \( 5m^{6}(2m^{7}+3m^{4}+m^{2}+m+1) \), simplification through the aforementioned steps results in the much simpler \( 10m^{13} + 15m^{10} + 5m^8 + 5m^7 + 5m^6 \). This expression is now ready for any further mathematical processing or analysis. Simplifying expressions is a vital skill that helps significantly when solving equations and understanding functional relationships in algebra.