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Chapter 6: Problem 39

Write with a single exponent. $$ B^{a} B^{a+1} $$

Short Answer

Expert verified
Question: Combine the given expressions \(B^a B^{a+1}\) into a single expression with a single exponent. Answer: \(B^{2a+1}\)

Step by step solution

01

Use the product of powers rule

To combine the expressions \(B^a B^{a+1}\), use the product of powers rule: \(B^{m} B^n = B^{m+n}\), where m and n are the exponents. In our case, m=a and n=a+1.
02

Add the exponents

Since m=a and n=a+1, we have: \(B^a B^{a+1} = B^{a+(a+1)}\)
03

Simplify the exponent

Now simplify the exponent: \(a + (a + 1) = a + a + 1 = 2a + 1\)
04

Write the final expression

Substitute the simplified exponent back into the expression: \(B^{a+(a+1)} = B^{2a+1}\) So, the expression \(B^a B^{a+1}\) written with a single exponent is: $$ B^{2a+1} $$

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Key Concepts

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

Product of Powers Rule
When working with exponents, there's a super useful rule known as the product of powers rule. This rule helps us combine terms with the same base by adding their exponents together. It's a neat way to simplify expressions!
To use this rule, you take two expressions where they have the same base, like \( B^m \) and \( B^n \), and simply add their exponents: \[ B^m \, B^n = B^{m+n} \].
For example, if we have \( B^a \, B^{a+1} \), you apply the product of powers rule by adding \(a\) and \(a+1\), giving you \( B^{a+(a+1)} \). Easy, right? By understanding the product of powers rule, you can simplify expressions in no time and turn them into a single term.
Simplifying Exponents
After you've applied the product of powers rule, the next step is simplifying exponents. This means you'll need to do some addition within the exponent itself.
So, if you started with \( B^{a+(a+1)} \), to simplify, you combine like terms in the exponent: \(a + (a + 1) \) becomes \( a + a + 1 \). This is straightforward!
By combining, you get \( 2a + 1 \). Always remember that simplifying exponents involves basic arithmetic, just performed with exponents rather than the typical integers.
Keep in mind that mastering simplification is a key skill in algebra because it allows you to express results more compactly and clearly.
Combining Like Terms
Combining like terms is another crucial skill when working with algebraic expressions. It's all about merging terms that have the same variable parts.
In our specific case, after applying the product of powers rule, we arrived at the exponent \( 2a + 1 \).
When combining terms in an exponent, look for identical variable parts, which simply means adding the coefficients of those parts.
  • Here, \(a\) and \(a\) in the expression \(a + a + 1\) are like terms. By adding them, you get \(2a\).
  • The \(+1\) is a constant that just comes along for the ride.
Understanding how to combine like terms efficiently will help you simplify processes and tackle more complex problems with confidence.

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