To determine which student correctly simplified the expression [tex]\(\frac{4^{10}}{4^5}\)[/tex], let's proceed with the following steps:
1. Understanding the Law of Exponents for Division:
- When we divide two expressions with the same base, we subtract the exponents.
- The general rule is: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
2. Applying the Law of Exponents:
- Given the expression [tex]\(\frac{4^{10}}{4^5}\)[/tex]:
[tex]\[
\frac{4^{10}}{4^5} = 4^{10-5}
\][/tex]
3. Simplifying the Expression:
- Subtract the exponents in the numerator and denominator:
[tex]\[
4^{10-5} = 4^5
\][/tex]
4. Evaluating the Simplified Expression:
- The next step is to compute the value of [tex]\(4^5\)[/tex].
[tex]\[
4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 1024
\][/tex]
By following the correct steps of simplifying the expression using the laws of exponents, we find the simplified expression [tex]\(4^5 = 1024\)[/tex].
5. Identifying the Correct Student:
- Camilla added the exponents: [tex]\(4^{10+5} = 4^{15}\)[/tex] — This is incorrect.
- Bentley divided the exponents: [tex]\(4^{10 \div 5} = 4^2\)[/tex] — This is incorrect.
- Mira multiplied the exponents: [tex]\(4^{10 \cdot 5} = 4^{50}\)[/tex] — This is incorrect.
- Regan subtracted the exponents: [tex]\(4^{10 - 5} = 4^5\)[/tex] — This is correct.
Therefore, the student who correctly simplified the expression [tex]\(\frac{4^{10}}{4^5}\)[/tex] is Regan.
