Identify Prime Factors: Identify the prime factors of 56 and express z^(7) as z^(6) * z to simplify the square root. 56 can be factored into 2 * 28, which further factors into 2 * 2 * 14, and finally into 2 * 2 * 2 * 7. So, 56 = 2^3 * 7. z^(7) can be written as z^(6) * z to separate an even exponent that can be simplified under the square root.Rewrite Product as Square Roots: Rewrite the square root of the product as the product of square roots. sqrt(56z^(7)) = sqrt(2^3 * 7 * z^(6) * z) = sqrt(2^3 * 7) * sqrt(z^(6)) * sqrt(z).Simplify Square Roots: Simplify the square roots. sqrt(2^3 * 7) simplifies to 2 * sqrt(7) because 2^2 (which is 4) is the largest square factor of 2^3. sqrt(z^(6)) simplifies to z^(3) because z^(6) is a perfect square (z^(6/2) = z^(3)). sqrt(z) remains as it is because z does not have an even exponent.Combine Simplified Roots: Combine the simplified square roots into a single expression. The simplified form is 2z^(3) * sqrt(7) * sqrt(z).Check Further Simplification: Check if the expression can be further simplified. Since there are no like terms or further square roots that can be simplified, this is the final simplified form.

Precalculus

Evaluate rational exponents

Full solution

\sqrt{56z^{7}}

Identify Prime Factors: Identify the prime factors of $56$ and express $z^{7}$ as $z^{6} \times z$ to simplify the square root. $\newline$ $56$ can be factored into $2 \times 28$ , which further factors into $2 \times 2 \times 14$ , and finally into $2 \times 2 \times 2 \times 7$ . So, $56 = 2^{3} \times 7$ . $\newline$ $z^{7}$ can be written as $z^{6} \times z$ to separate an even exponent that can be simplified under the square root.
Rewrite Product as Square Roots: Rewrite the square root of the product as the product of square roots. $\sqrt{56z^{7}} = \sqrt{2^3 \times 7 \times z^{6} \times z} = \sqrt{2^3 \times 7} \times \sqrt{z^{6}} \times \sqrt{z}$ .
Simplify Square Roots: Simplify the square roots. $\sqrt{2^3 \times 7}$ simplifies to $2 \times \sqrt{7}$ because $2^2$ (which is $4$ ) is the largest square factor of $2^3$ . $\sqrt{z^{6}}$ simplifies to $z^{3}$ because $z^{6}$ is a perfect square ( $z^{6/2} = z^{3}$ ). $\sqrt{z}$ remains as it is because $2 \times \sqrt{7}$ $0$ does not have an even exponent.
Combine Simplified Roots: Combine the simplified square roots into a single expression. $\newline$ The simplified form is $2z^{3} \times \sqrt{7} \times \sqrt{z}$ .
Check Further Simplification: Check if the expression can be further simplified. $\newline$ Since there are no like terms or further square roots that can be simplified, this is the final simplified form.