Problem 71 - Algebra 1: Concepts and Skills

Question

Simplify the expression. Write your answer as a power. $$ b^{2} \cdot b^{5} $$

Step-by-Step Solution

Answer

The simplified expression is \( b^{7} \)

Step 1: Understanding the problem

This problem can be solved by applying the rule, which states that\( a^{m} \cdot a^{n} = a^{m+n} \), where 'a' is the base, and 'm' and 'n' are the exponents.

Step 2: Applying the rule

Here \( b^{2} \cdot b^{5} = b^{2+5} \). We sum the exponents 2 and 5.

Step 3: Simplify the expression

The summation of 2 and 5 is 7. Therefore, \( b^{2} \cdot b^{5} = b^{7}. \)

Key Concepts

Algebra

Algebra is a crucial part of mathematics that deals with symbols and the rules for manipulating those symbols to express mathematical ideas and relationships in a form that can be easily worked with. Instead of using numbers directly, algebra uses letters, like 'b' in our given expression, to represent unknown or general numbers. This allows us to create formulas and equations that can be used to generalize and solve problems.
In our problem, the symbol 'b' is used. Understanding algebra helps us utilize the vast set of mathematical tools to rearrange and solve expressions and equations. The problem we've looked at is an example of using algebraic rules to handle expressions with exponents.

Laws of Exponents

The laws of exponents are a set of rules that simplify the manipulation of algebraic expressions containing powers. Here are the basic laws:

• Product of Powers Rule: This states that when you multiply two powers with the same base, you can add their exponents: \( a^{m} \cdot a^{n} = a^{m+n} \).
• Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^{m})^{n} = a^{m \cdot n} \).
• Power of a Product Rule: When raising a product to a power, you can apply the exponent to each factor: \((ab)^{n} = a^{n}b^{n}\).

In our problem, we used the Product of Powers Rule to simplify the expression from \( b^{2} \cdot b^{5} \) into \( b^{7} \) by adding the exponents. Understanding these laws allows us to simplify and solve expressions much more efficiently.

Simplifying Expressions

Simplifying expressions involves reducing them to a form that's easier to work with and understand, without changing their value. The goal is to make expressions as simple as possible.
In algebra, this can mean combining like terms or reducing powers with the help of the laws of exponents. For example, simplifying \( b^{2} \cdot b^{5} \) means applying those rules to reduce the expression to \( b^{7} \). This kind of simplification is crucial because it helps to solve equations more easily and find clear, concise forms of expressions. It's a vital skill in mathematics that makes solving equations and understanding complex problems much less intimidating.
By recursively applying algebraic rules and understanding the fundamental properties of powers and numbers, we can ensure our mathematical expressions are always in their simplest and most manageable form.