To determine if each expression is in simplest form, we need to simplify each one step by step.
1. Expression: [tex]\( a \sqrt{60b} \)[/tex]
Let's simplify [tex]\( \sqrt{60b} \)[/tex]:
[tex]\[
\sqrt{60b} = \sqrt{4 \cdot 15 \cdot b} = \sqrt{4} \cdot \sqrt{15b} = 2\sqrt{15b}
\][/tex]
Therefore,
[tex]\[
a \sqrt{60b} = a \cdot 2\sqrt{15b} = 2a\sqrt{15b}
\][/tex]
This expression can be simplified to [tex]\( 2a\sqrt{15b} \)[/tex]. Thus, [tex]\( a \sqrt{60b} \)[/tex] is not in simplest form.
2. Expression: [tex]\( 49 \sqrt{a} \)[/tex]
This expression cannot be simplified further because there are no factors inside the square root that can be simplified. Therefore, it is in simplest form. Thus, [tex]\( 49 \sqrt{a} \)[/tex] is in simplest form.
3. Expression: [tex]\( 9 a^2 b \sqrt{5ab} \)[/tex]
Let's simplify [tex]\( \sqrt{5ab} \)[/tex]:
[tex]\[
\sqrt{5ab}
\][/tex]
There are no factors to be simplified further inside the square root. Therefore,
[tex]\[
9 a^2 b \sqrt{5ab}
\][/tex]
This expression cannot be simplified further either. Thus, [tex]\( 9 a^2 b \sqrt{5ab} \)[/tex] is in simplest form.
4. Expression: [tex]\( \sqrt{28 a^3 b^8} \)[/tex]
Let's simplify [tex]\( \sqrt{28 a^3 b^8} \)[/tex]:
[tex]\[
\sqrt{28 a^3 b^8} = \sqrt{4 \cdot 7 \cdot a^3 \cdot b^8} = \sqrt{4} \cdot \sqrt{7} \cdot \sqrt{a^3} \cdot \sqrt{b^8} = 2\sqrt{7} \cdot a^{3/2} \cdot b^4 = 2b^4a^{3/2}\sqrt{7}
\][/tex]
Therefore, [tex]\( \sqrt{28 a^3 b^8} \)[/tex] simplifies to [tex]\( 2a^{3/2} b^4 \sqrt{7} \)[/tex]. Thus, [tex]\( \sqrt{28 a^3 b^8} \)[/tex] is not in simplest form.
5. Expression: [tex]\( 4 a^2 \sqrt{b^3} \)[/tex]
Let's simplify [tex]\( \sqrt{b^3} \)[/tex]:
[tex]\[
\sqrt{b^3} = \sqrt{b^2 \cdot b} = \sqrt{b^2} \cdot \sqrt{b} = b \sqrt{b}
\][/tex]
Therefore,
[tex]\[
4 a^2 \sqrt{b^3} = 4 a^2 b \sqrt{b}
\][/tex]
This expression simplifies to [tex]\( 4a^2 b \sqrt{b} \)[/tex]. Thus, [tex]\( 4 a^2 \sqrt{b^3} \)[/tex] is not in simplest form.
Based on this analysis:
Correct simplest form expressions:
- [tex]\( 49 \sqrt{a} \)[/tex]
- [tex]\( 9 a^2 b \sqrt{5ab} \)[/tex]
