Problem 101 - Elementary Algebra

Question

Simplify the following problems. $$ 2 b^{5} 2 b^{3} $$

Step-by-Step Solution

Answer

Answer: \(4b^8\).

Step 1: Identify the like terms

We are given the expression \(2b^5 \cdot 2b^3\). Notice that both terms have the same base b and the coefficients are both multiples of 2.

Step 2: Multiply the coefficients

The coefficients of the two terms are 2 and 2. When multiplying the terms, we should multiply the coefficients together: \(2 \cdot 2 = 4\).

Step 3: Apply exponent rules for multiplication

Since both terms have a base of b, we can add their exponents when multiplying. So, we have: \(b^5 \cdot b^3 = b^{5+3} = b^8\).

Step 4: Combine the coefficient and base

Finally, we can combine the coefficient we found in Step 2 and the base we found in Step 3 to get our final simplified expression: \(4b^8\). So, the simplified expression is \(4b^8\).

Key Concepts

Exponent Rules

When working with algebraic expressions that involve exponents, it's crucial to understand the basics of exponent rules. In simple terms, exponents are a way to denote repeated multiplication. For instance, when we see something like \(b^5\), it means \(b\) multiplied by itself five times: \(b \cdot b \cdot b \cdot b \cdot b\).

The most common rule you'll encounter is when you're multiplying terms that share the same base, such as \(b^5\) and \(b^3\). Instead of multiplying the numbers directly, you can add their exponents together as their bases are the same. This is expressed as \(b^m \cdot b^n = b^{m+n}\). So, in our example, \(b^5 \cdot b^3 = b^{5+3} = b^8\).

• Multiplying bases: Keep the base the same, add the exponents.
• Division of bases: Keep the base, subtract the exponents.
• Power of a power: Multiply the exponents.

Understanding these rules will help simplify complex algebraic expressions immediately.

Multiplication of Terms

Multiplying algebraic terms involves both the coefficients (numerical part) and the variables (letters with exponents). Taking a step-by-step approach can make this process much easier. Consider the expression \(2b^5 \cdot 2b^3\).

First, focus on the coefficients—these are the numbers in front of any variables. In this case, we have \(2\) and \(2\). To multiply them, simply multiply these numbers together, \(2 \cdot 2 = 4\).

• Always multiply numerical coefficients separately from variable parts.
• Ensure that you're applying the correct operation—sometimes expressions may require division, not multiplication.

Once coefficients are handled, apply the exponent rules to the variables as discussed. Remember to handle each part of the term separately for precision.

Combining Like Terms

Combining like terms is a key concept in simplifying expressions. It's all about putting together terms that have identical variable parts. In the expression \(2b^5 \cdot 2b^3\), even though we're multiplying and not directly combining terms, a similar idea of recognizing alike elements applies.

After applying exponent rules and multiplying coefficients, the expression becomes \(4b^8\). Unlike addition or subtraction where you combine terms directly, multiplication needs consistent bases to modify exponents.

• Identify terms with the same variable parts and exponents.]
• Keep the coefficient section and variable section as separate parts to easily manage larger expressions.

By understanding the underlying similarities, such as recognizing consistent variables and their exponents, you're better equipped to simplify complex algebraic problems.