https://file.aiquickdraw.com/mathbot/file/1725975367740os1yfztx.jpg

The expression to simplify is:

\[
\sqrt{56z^7}
\]

To simplify this, we need to remove all perfect squares from inside the square root.

### Step 1: Factor 56 into its prime factors.
\[
56 = 2^3 \times 7
\]

### Step 2: Break down the variable \( z^7 \) as powers of perfect squares.
We can express \( z^7 \) as:
\[
z^7 = z^6 \times z = (z^3)^2 \times z
\]

### Step 3: Apply the square root property.
We know that:
\[
\sqrt{a^2 \times b} = a \times \sqrt{b}
\]

Thus:
\[
\sqrt{56z^7} = \sqrt{2^3 \times 7 \times (z^3)^2 \times z}
\]

### Step 4: Simplify the expression.
- The perfect square part from \( z^7 \) is \( (z^3)^2 \), which becomes \( z^3 \) outside the square root.
- \( 2^3 \) can be split as \( 2^2 \times 2 \), so \( 2^2 \) comes out as 2.
- The remaining terms inside the square root are \( 2 \times 7 \times z = 14z \).

So, the simplified expression is:
\[
2z^3 \sqrt{14z}
\]

Thus:
\[
\sqrt{56z^7} = 2z^3 \sqrt{14z}
\]

Would you like more details on any of these steps or have questions? Here are some related questions to consider:

1. How do you simplify square roots involving higher powers of variables?
2. How do you factor numbers like 56 into primes?
3. What are some common perfect squares and cubes to recognize?
4. Can any non-perfect square factors ever fully be removed from a square root?
5. How would you handle fractional exponents for square roots?

**Tip:** When simplifying square roots, always look for perfect squares in both the numerical and variable components for easier simplification.

Simplify. Remove all perfect squares from inside the square root: √(56z^7) = ?

Learn how to simplify the square root of 56z^7 by removing perfect squares. This detailed guide covers square roots, exponent rules, and factoring integers. Suitable for students in high school.

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