Problem 591
Question
In the following exercises, simplify. $$ n^{\frac{2}{6}} \cdot n^{\frac{4}{6}} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \( n \).
1Step 1: Identify the Base and Exponents
Recognize that both terms have the same base, which is the variable \(n\). The exponents are \(\frac{2}{6}\) and \(\frac{4}{6}\).
2Step 2: Apply the Product of Powers Rule
When multiplying expressions with the same base, add the exponents. Apply the rule: \( n^a \cdot n^b = n^{a+b} \). Here, \( a = \frac{2}{6} \) and \( b = \frac{4}{6} \).
3Step 3: Add the Exponents
Add the exponents: \( \frac{2}{6} + \frac{4}{6} = \frac{2+4}{6} = \frac{6}{6} \).
4Step 4: Simplify the Exponent
Simplify the exponent: \( \frac{6}{6} = 1 \).
5Step 5: Write the Simplified Expression
The simplified expression is: \( n^1 \), which can also be written as just \( n \).
Key Concepts
product of powers ruleadding exponentssimplified expression
product of powers rule
When you come across an expression where you multiply terms that have the same base, you can simplify it easily using the product of powers rule. The rule states that when you multiply two powers with the same base, you add their exponents.
For example, if you have \(n^a \cdot n^b\), you can simplify it to \(n^{a+b}\). This works because you are essentially adding the number of times the base is multiplied by itself.
So, in the given exercise, when you have \(n^{\frac{2}{6}}\cdot n^{\frac{4}{6}}\), you just add the exponents: \(\frac{2}{6} + \frac{4}{6}\).
For example, if you have \(n^a \cdot n^b\), you can simplify it to \(n^{a+b}\). This works because you are essentially adding the number of times the base is multiplied by itself.
So, in the given exercise, when you have \(n^{\frac{2}{6}}\cdot n^{\frac{4}{6}}\), you just add the exponents: \(\frac{2}{6} + \frac{4}{6}\).
adding exponents
Adding exponents occurs when you use the product of powers rule. The critical thing to remember is that this rule only applies when the bases are the same.
In the exercise, we have our exponents \(\frac{2}{6}\) and \(\frac{4}{6}\).
When we sum these, we get: \(\frac{2}{6} + \frac{4}{6} = \frac{2+4}{6} = \frac{6}{6}\).
Adding these fractions is simple by having a common denominator, which in this case is 6. So, the sum of the exponents simplifies to \(\frac{6}{6}\), which is just 1.
In the exercise, we have our exponents \(\frac{2}{6}\) and \(\frac{4}{6}\).
When we sum these, we get: \(\frac{2}{6} + \frac{4}{6} = \frac{2+4}{6} = \frac{6}{6}\).
Adding these fractions is simple by having a common denominator, which in this case is 6. So, the sum of the exponents simplifies to \(\frac{6}{6}\), which is just 1.
simplified expression
The final step in the exercise is to present the simplified expression.
After following through the steps of identifying the base, using the product of powers rule, and adding the exponents, we need to simplify the resulting exponent.
From the previous steps, we found that \(n^{\frac{6}{6}}\) simplifies to \(n^1\), and any number or variable raised to the power of 1 is itself.
So, the simplified expression of the exercise \( n^{\frac{2}{6}} \cdot n^{\frac{4}{6}} \) is simply \(n\). This means that multiplying these two expressions results in the same base, but only appearing once, to the power of 1.
This not only simplifies your work but also makes understanding and handling these expressions much easier going forward!
After following through the steps of identifying the base, using the product of powers rule, and adding the exponents, we need to simplify the resulting exponent.
From the previous steps, we found that \(n^{\frac{6}{6}}\) simplifies to \(n^1\), and any number or variable raised to the power of 1 is itself.
So, the simplified expression of the exercise \( n^{\frac{2}{6}} \cdot n^{\frac{4}{6}} \) is simply \(n\). This means that multiplying these two expressions results in the same base, but only appearing once, to the power of 1.
This not only simplifies your work but also makes understanding and handling these expressions much easier going forward!
Other exercises in this chapter
Problem 589
In the following exercises, simplify. (a) \(\frac{m^{\frac{7}{4}} \cdot m^{-\frac{5}{4}}}{m^{-\frac{2}{4}}}\) (b) \(\frac{n^{\frac{3}{7}} \cdot n}{n^{-\frac{4}{
View solution Problem 590
In the following exercises, simplify. $$ 4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}} $$
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In the following exercises, simplify. $$ \left(a^{24}\right)^{\frac{1}{6}} $$
View solution Problem 593
In the following exercises, simplify. $$ \left(b^{10}\right)^{\frac{3}{5}} $$
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