Asked by Alto Saxaphone
10^4+10^6 keep in exponent form
Answers
Answered by
Alto Saxaphone
how did you get your answer
Answered by
GPT-4o mini
To keep the expression \(10^4 + 10^6\) in exponent form, you can factor out the smaller exponent, which is \(10^4\):
\[
10^4 + 10^6 = 10^4(1 + 10^{6-4}) = 10^4(1 + 10^2)
\]
So, the expression in exponent form is:
\[
10^4(1 + 10^2)
\]
You can also simplify \(1 + 10^2\):
\[
= 10^4(1 + 100) = 10^4 \cdot 101
\]
Thus, the final expression can be presented as:
\[
10^4 \cdot 101
\]
However, if you're strictly looking to maintain the exponent form without evaluating \(1 + 10^2\) as a numeric factor, then the first factored form \(10^4(1 + 10^2)\) is the best representation in exponent form.
\[
10^4 + 10^6 = 10^4(1 + 10^{6-4}) = 10^4(1 + 10^2)
\]
So, the expression in exponent form is:
\[
10^4(1 + 10^2)
\]
You can also simplify \(1 + 10^2\):
\[
= 10^4(1 + 100) = 10^4 \cdot 101
\]
Thus, the final expression can be presented as:
\[
10^4 \cdot 101
\]
However, if you're strictly looking to maintain the exponent form without evaluating \(1 + 10^2\) as a numeric factor, then the first factored form \(10^4(1 + 10^2)\) is the best representation in exponent form.
Answered by
GPT-4o mini
To simplify the expression \(10^4 + 10^6\) and keep it in exponent form, I used the following steps:
1. **Identify the Terms**: The expression consists of two terms, \(10^4\) and \(10^6\).
2. **Factor Out the Smaller Exponent**: It's often useful to factor out the common base raised to the smaller exponent. Here, \(10^4\) is smaller than \(10^6\). So, I factor out \(10^4\):
\[
10^4 + 10^6 = 10^4(1 + \frac{10^6}{10^4})
\]
3. **Simplify the Fraction**: When you divide \(10^6\) by \(10^4\), you can subtract the exponents:
\[
\frac{10^6}{10^4} = 10^{6-4} = 10^2
\]
Substituting this back into the expression gives:
\[
10^4(1 + 10^2)
\]
4. **Simplify Further**: At this point, you can either leave it as \(10^4(1 + 10^2)\) or, if desired, evaluate \(1 + 10^2\):
\[
1 + 10^2 = 1 + 100 = 101
\]
Thus, you can express it as:
\[
10^4 \cdot 101
\]
5. **Final Expression**: The result keeps the base \(10\) factored out, and in this form, it clearly shows the exponent related to the term \(10^4\), while \(101\) is a separate coefficient.
In summary, the main goal was to factor the expression in a way that keeps the powers of 10 together, which allows for a clearer representation of the equation in exponent form. Thus the final answer could be expressed as either:
\[
10^4(1 + 10^2) \quad \text{or} \quad 10^4 \cdot 101
\]
depending on your requirements.
1. **Identify the Terms**: The expression consists of two terms, \(10^4\) and \(10^6\).
2. **Factor Out the Smaller Exponent**: It's often useful to factor out the common base raised to the smaller exponent. Here, \(10^4\) is smaller than \(10^6\). So, I factor out \(10^4\):
\[
10^4 + 10^6 = 10^4(1 + \frac{10^6}{10^4})
\]
3. **Simplify the Fraction**: When you divide \(10^6\) by \(10^4\), you can subtract the exponents:
\[
\frac{10^6}{10^4} = 10^{6-4} = 10^2
\]
Substituting this back into the expression gives:
\[
10^4(1 + 10^2)
\]
4. **Simplify Further**: At this point, you can either leave it as \(10^4(1 + 10^2)\) or, if desired, evaluate \(1 + 10^2\):
\[
1 + 10^2 = 1 + 100 = 101
\]
Thus, you can express it as:
\[
10^4 \cdot 101
\]
5. **Final Expression**: The result keeps the base \(10\) factored out, and in this form, it clearly shows the exponent related to the term \(10^4\), while \(101\) is a separate coefficient.
In summary, the main goal was to factor the expression in a way that keeps the powers of 10 together, which allows for a clearer representation of the equation in exponent form. Thus the final answer could be expressed as either:
\[
10^4(1 + 10^2) \quad \text{or} \quad 10^4 \cdot 101
\]
depending on your requirements.
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