Factorize and Simplify: First, we need to simplify the square root of the product sqrt(72x^3z^3). To do this, we will factor 72 into its prime factors and look for pairs of factors and variables under the square root since sqrt(a^2) = a. 72 can be factored into 2 * 36, and 36 can be further factored into 2 * 18, and 18 can be factored into 2 * 9, and 9 can be factored into 3 * 3. So, the prime factorization of 72 is 2 * 2 * 2 * 3 * 3, or 2^3 * 3^2. For the variables, we have x^3 and z^3. Since we are taking the square root, we will look for pairs of x's and z's.Rewrite Using Prime Factorization: Now, we will rewrite the square root of the product using the prime factorization and the variables: sqrt(72x^3z^3) = sqrt(2^3 * 3^2 * x^3 * z^3). We can pair up the factors under the square root to simplify: sqrt(2^2 * 2 * 3^2 * x^2 * x * z^2 * z).Take Out Pairs: Next, we take out the pairs from under the square root, remembering that the square root of a square is just the base: sqrt(2^2) * sqrt(3^2) * sqrt(x^2) * sqrt(z^2) = 2 * 3 * x * z. We are left with the unpaired factors under the square root: sqrt(2 * x * z).Combine and Simplify: Combining the terms we took out of the square root with the terms left inside, we get: 2 * 3 * x * z * sqrt(2 * x * z) = 6xz * sqrt(2xz).Final Simplified Expression: Finally, we write the simplified expression for sqrt(72x^3z^3): sqrt(72x^3z^3) = 6xz * sqrt(2xz).

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Algebra 2

Csc, sec, and cot of special angles

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\sqrt{72x^{3}z^{3}}=

Factorize and Simplify: First, we need to simplify the square root of the product $\sqrt{72x^3z^3}$ . To do this, we will factor $72$ into its prime factors and look for pairs of factors and variables under the square root since $\sqrt{a^2} = a$ . $\newline$ $72$ can be factored into $2 \times 36$ , and $36$ can be further factored into $2 \times 18$ , and $18$ can be factored into $2 \times 9$ , and $9$ can be factored into $72$ $0$ . So, the prime factorization of $72$ is $72$ $2$ , or $72$ $3$ . $\newline$ For the variables, we have $72$ $4$ and $72$ $5$ . Since we are taking the square root, we will look for pairs of $72$ $6$ 's and $72$ $7$ 's.
Rewrite Using Prime Factorization: Now, we will rewrite the square root of the product using the prime factorization and the variables: $\newline$ $\sqrt{72x^3z^3} = \sqrt{2^3 \cdot 3^2 \cdot x^3 \cdot z^3}$ . $\newline$ We can pair up the factors under the square root to simplify: $\newline$ $\sqrt{2^2 \cdot 2 \cdot 3^2 \cdot x^2 \cdot x \cdot z^2 \cdot z}$ .
Take Out Pairs: Next, we take out the pairs from under the square root, remembering that the square root of a square is just the base: $\sqrt{2^2} \times \sqrt{3^2} \times \sqrt{x^2} \times \sqrt{z^2} = 2 \times 3 \times x \times z$ . We are left with the unpaired factors under the square root: $\sqrt{2 \times x \times z}$ .
Combine and Simplify: Combining the terms we took out of the square root with the terms left inside, we get: $\newline$ $2 \times 3 \times x \times z \times \sqrt{2 \times x \times z} = 6xz \times \sqrt{2xz}$ .
Final Simplified Expression: Finally, we write the simplified expression for $\sqrt{72x^3z^3}$ : $\sqrt{72x^3z^3} = 6xz \times \sqrt{2xz}$ .