Question

1. simplify.

$sqrt{sqrt3{3^6}}$
options: 3, $3^6$, $3^2$, $3^3$

1. simplify.

$sqrt3{64x^3y^6}$
options: $2xy^2$, $4xy^2$, $(2xy)^2$, $16xy^2$

1. simplify.

$left( sqrt{5x^3}
ight)^2$
options: $sqrt3{5x^2}$, $5x^6$, $sqrt{5x}$, $5x^3$

Explanation:

Problem 7:

Step1: Rewrite roots as exponents

The expression is $\sqrt{\sqrt[3]{3^6}}$. First, recall that $\sqrt[n]{a}=a^{\frac{1}{n}}$. So, $\sqrt[3]{3^6}=(3^6)^{\frac{1}{3}}$.

Step2: Simplify the inner exponent

Using the power - of - a - power rule $(a^m)^n=a^{m\times n}$, we have $(3^6)^{\frac{1}{3}} = 3^{6\times\frac{1}{3}}=3^2$.

Step3: Simplify the outer square root

Now, we have $\sqrt{3^2}$. Since $\sqrt{a^2}=|a|$, and for $a = 3$ (a positive number), $\sqrt{3^2}=3$.

Step1: Rewrite the cube root as an exponent

The expression is $\sqrt[3]{64x^3y^6}=(64x^3y^6)^{\frac{1}{3}}$.

Step2: Apply the power - of - a - product rule

Using the rule $(ab)^n=a^n\times b^n$, we get $(64x^3y^6)^{\frac{1}{3}}=64^{\frac{1}{3}}\times(x^3)^{\frac{1}{3}}\times(y^6)^{\frac{1}{3}}$.

Step3: Simplify each term

We know that $64 = 4^3$, so $64^{\frac{1}{3}}=(4^3)^{\frac{1}{3}} = 4$. Using the power - of - a - power rule $(a^m)^n=a^{m\times n}$, $(x^3)^{\frac{1}{3}}=x^{3\times\frac{1}{3}}=x$ and $(y^6)^{\frac{1}{3}}=y^{6\times\frac{1}{3}}=y^2$.

Step4: Multiply the simplified terms

Multiplying these together, we get $4\times x\times y^2 = 4xy^2$.

Step1: Apply the power - of - a - square - root rule

The expression is $(\sqrt{bx^3})^2$. Recall that $(\sqrt{a})^2=a$ for $a\geq0$. So, $(\sqrt{bx^3})^2 = bx^3$. Wait, there seems to be a typo in the problem (maybe it's $(\sqrt{5x^3})^2$ instead of $(\sqrt{bx^3})^2$). Assuming it's $(\sqrt{5x^3})^2$:

Step2: Simplify using the rule $(\sqrt{a})^2=a$

Using the rule $(\sqrt{a})^2=a$ (where $a = 5x^3$), we get $5x^3$.

Answer:

3 (the first option)

Problem 8: