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Chapter 1: Problem 5

Simplify the given expressions. Express results with positive exponents only. $$2 b^{4} b^{2}$$

Short Answer

Expert verified
The simplified expression is \\(2 b^6\\).

Step by step solution

01

Combine the Exponents

First, observe that both the terms have the base \(b\). We can simplify \(b^4 b^2\) by using the property of exponents: \(a^m a^n = a^{m+n}\). So, we add the exponents of \(b\): \(4 + 2 = 6\). Thus, \(b^4 b^2 = b^6\).
02

Simplify the Expression

Now, rewrite the original expression by substituting the simplified exponent from the previous step: \(2 b^4 b^2 = 2 b^6\). All exponents are already positive, so nothing further is needed in terms of adjusting signs.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a fundamental concept in algebra and mathematics in general. An exponent tells us how many times a number, known as the base, is multiplied by itself. In the expression \(b^n\), \(b\) is the base, and \(n\) is the exponent. For example, \(b^4\) means that the base \(b\) is multiplied by itself four times: \(b \times b \times b \times b\). Exponents allow us to write lengthy multiplication processes in a concise way.

Understanding how exponents work is vital as they are widely used in science, engineering, and mathematics.
  • They simplify expressions and make calculations easier.
  • They allow for the representation of large numbers compactly.
  • They are foundational for understanding more advanced topics, like polynomials and logarithms.
In the exercise, simplifying \(2b^4b^2\) requires applying the properties of exponents to combine terms with the same base.
Combining Like Terms
Combining like terms is a process that involves simplifying expressions by adding or subtracting terms that have the same variables raised to the same power. This process is key in solving algebraic equations and simplifying expressions for easier calculation.

To combine like terms, follow these steps:
  • Identify terms with the same variable raised to the same power. These are your like terms.
  • Add or subtract the coefficients (the numbers in front of the variables). The variable part remains unchanged.
  • Repeat this as necessary to simplify the entire expression.
In our original exercise \(2b^4b^2\), the terms \(b^4\) and \(b^2\) share the common base \(b\).
Add their exponents to simplify, which is a part of combining like terms when exponents are present: \(b^{4+2} = b^6\). Finally, the coefficient \(2\) remains separate, producing the simplified expression \(2b^6\).
Properties of Exponents
The properties of exponents are essential tools for simplifying expressions that involve powers. These properties help us manipulate and rewrite expressions so they become easier to understand and solve.

Some key properties include:
  • Product of Powers Property: This property states that when multiplying two exponents with the same base, you can simply add the exponents: \(a^m \times a^n = a^{m+n}\).
  • Power of a Power Property: When raising an exponent to another power, multiply the exponents: \((a^m)^n = a^{m\times n}\).
  • Power of a Product Property: When raising a product to a power, apply the exponent to each factor in the product: \((ab)^n = a^n \times b^n\).
In our original exercise, we focused on the "Product of Powers Property" to combine \(b^4\) and \(b^2\). Since their bases are the same, we added their exponents, yielding \(b^6\). Recognizing these properties can significantly reduce the time and effort needed to simplify mathematical expressions.

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Most popular questions from this chapter

Perform the calculations on a calculator without rounding. Evaluate: (a) \(6.03 \div 2.25+1.77 \quad\) (b) \(6.03 \div(2.25+1.77)\)

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