Question

question
rewrite the following expression in simplest radical form given ( x > 0 ).
( sqrt4{16x^8} cdot sqrt{9x^5} )

Explanation:

Step1: Simplify the fourth root

Simplify $\sqrt[4]{16x^8}$. We know that $16 = 2^4$ and $x^8=(x^2)^4$. So, $\sqrt[4]{16x^8}=\sqrt[4]{2^4\cdot(x^2)^4}=2x^2$ (since $x>0$, we don't have to consider absolute values).

Step2: Simplify the square root

Simplify $\sqrt{9x^5}$. We know that $9 = 3^2$ and $x^5=x^4\cdot x$. So, $\sqrt{9x^5}=\sqrt{3^2\cdot x^4\cdot x}=3x^2\sqrt{x}$ (again, $x>0$ so we can take the square root of $x^4$ as $x^2$).

Step3: Multiply the simplified expressions

Now, multiply the results from Step 1 and Step 2: $(2x^2)\cdot(3x^2\sqrt{x})$. First, multiply the coefficients: $2\times3 = 6$. Then, multiply the variables with the same base using the rule $a^m\cdot a^n=a^{m + n}$: $x^2\cdot x^2=x^{2 + 2}=x^4$. So, putting it together, we have $6x^4\sqrt{x}$. Wait, but let's check again. Wait, maybe we can also use exponent rules from the start by converting radicals to exponents. Let's try that method to verify.

Alternative Step1: Convert radicals to exponents
Recall that $\sqrt[n]{a}=a^{\frac{1}{n}}$. So, $\sqrt[4]{16x^8}=(16x^8)^{\frac{1}{4}}$ and $\sqrt{9x^5}=(9x^5)^{\frac{1}{2}}$.

Alternative Step2: Simplify the exponent expressions
Simplify $(16x^8)^{\frac{1}{4}}$: Using the power of a product rule $(ab)^n=a^n b^n$ and power of a power rule $(a^m)^n=a^{mn}$, we get $16^{\frac{1}{4}}\cdot(x^8)^{\frac{1}{4}}$. Since $16 = 2^4$, $16^{\frac{1}{4}}=(2^4)^{\frac{1}{4}}=2$. And $(x^8)^{\frac{1}{4}}=x^{8\times\frac{1}{4}}=x^2$. So, $(16x^8)^{\frac{1}{4}} = 2x^2$.

Simplify $(9x^5)^{\frac{1}{2}}$: Similarly, $9^{\frac{1}{2}}\cdot(x^5)^{\frac{1}{2}}$. Since $9 = 3^2$, $9^{\frac{1}{2}}=(3^2)^{\frac{1}{2}}=3$. And $(x^5)^{\frac{1}{2}}=x^{\frac{5}{2}}$. So, $(9x^5)^{\frac{1}{2}} = 3x^{\frac{5}{2}}$.

Alternative Step3: Multiply the exponent expressions
Now, multiply $2x^2$ and $3x^{\frac{5}{2}}$. Multiply the coefficients: $2\times3 = 6$. Multiply the variables: $x^2\cdot x^{\frac{5}{2}}=x^{2+\frac{5}{2}}=x^{\frac{4 + 5}{2}}=x^{\frac{9}{2}}$.

Alternative Step4: Convert back to radical form
Recall that $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. So, $x^{\frac{9}{2}}=x^{4+\frac{1}{2}}=x^4\cdot x^{\frac{1}{2}}=x^4\sqrt{x}$. Then, multiply by the coefficient 6: $6x^4\sqrt{x}$. Wait, but let's check the first method again. Wait, in Step 1, we had $2x^2$, in Step 2, $3x^2\sqrt{x}$. Multiplying them: $2x^2\times3x^2\sqrt{x}=6x^{2 + 2}\sqrt{x}=6x^4\sqrt{x}$. That's the same result. Alternatively, we can also combine the radicals first using exponent rules. Let's try that.

Another Approach Step1: Express both radicals with the same index (optional, but let's see)
The first radical has index 4, the second has index 2. The least common multiple of 4 and 2 is 4. So, rewrite $\sqrt{9x^5}$ as a fourth root. We know that $\sqrt{9x^5}=(9x^5)^{\frac{1}{2}}=(9x^5)^{\frac{2}{4}}=\sqrt[4]{(9x^5)^2}=\sqrt[4]{81x^{10}}$. Then, the original expression is $\sqrt[4]{16x^8}\cdot\sqrt[4]{81x^{10}}=\sqrt[4]{16x^8\cdot81x^{10}}$.

Another Approach Step2: Multiply the radicands
Multiply $16x^8$ and $81x^{10}$: $16\times81 = 1296$, and $x^8\cdot x^{10}=x^{18}$. So, we have $\sqrt[4]{1296x^{18}}$.

Another Approach Step3: Simplify the fourth root
Factor 1296 and $x^{18}$ into perfect fourth powers. $1296 = 6^4$ (since $6^4 = 1296$), and $x^{18}=x^{16}\cdot x^2=(x^4)^4\cdot x^2$. So, $\sqrt[4]{6^4\cdot(x^4)^4\cdot x^2}=6x^4\sqrt[4]{x^2}$. But $\sqrt[4]{x^2}=\sqrt{x}$ (since $x>0$, $\sqrt[4]{x^2}=x^{\frac{2}{4}}=x^{\frac{1}{2}}=\sqrt{x}$). So, $\sqrt[4]{x^2}=\sqrt{x}$, so we get $6x^4\sqrt{x}$, which is the same result. So, all methods lead to $6x^4\sqrt{x}$. Wait, but let's check the calculation of $x^{18}$ when we multiplied $x^8$ and $x^{10}$: $8 + 10 = 18$, correct. Then, $x^{18}$ divided by $x^{16}$ (which is a perfect fourth power) is $x^2$, correct. And $1296 = 6^4$, correct. So, $\sqrt[4]{6^4\cdot(x^4)^4\cdot x^2}=6x^4\sqrt[4]{x^2}=6x^4\sqrt{x}$ (since $\sqrt[4]{x^2}=\sqrt{x}$ for $x>0$). So, all methods confirm that the simplified form is $6x^4\sqrt{x}$.

Answer:

$6x^4\sqrt{x}$