Question

apply the properties of exponents to answer 11 - 13. 11. three students wrote the following equality statements. circle the name of any student who is correct. caleb: $(x^{4})^{3}=(x^{3})^{4}$; ellie: $13^{4}\times13^{7}=(13^{4})^{7}$; florian: $\frac{y^{8}cdot y}{y^{2}}=(y^{2})^{3}$

Explanation:

Step1: Recall power - of - a - power property

The power - of - a - power property states that \((a^m)^n=a^{mn}\). For Caleb's statement \((x^4)^3=x^{4\times3}=x^{12}\) and \((x^3)^4=x^{3\times4}=x^{12}\), so Caleb is correct.

Step2: Recall product of powers property

The product of powers property states that \(a^m\times a^n=a^{m + n}\). So \(13^4\times13^7=13^{4 + 7}=13^{11}\), and \((13^4)^7=13^{4\times7}=13^{28}\). So Ellie is incorrect.

Step3: Recall product and quotient of powers properties

First, using the product of powers property \(y^8\cdot y=y^{8 + 1}=y^9\), and using the quotient of powers property \(\frac{y^9}{y^2}=y^{9-2}=y^7\). Also, \((y^2)^3=y^{2\times3}=y^6\). So Florian is incorrect.

Answer:

Caleb is correct.