Simplify 7 square root of 3yz*4 square root of 24yz

Solution:

Step 1: Decompose the number under the radical

Express $24yz$ as $2^2 \cdot (6yz)$.

Step 1.1: Extract the square factor from $24$

$7\sqrt{3yz} \cdot (4\sqrt{4(6yz)})$

Step 1.2: Represent $4$ as $2^2$

$7\sqrt{3yz} \cdot (4\sqrt{2^2 \cdot 6yz})$

Step 1.3: Insert parentheses for clarity

$7\sqrt{3yz} \cdot (4\sqrt{2^2 \cdot (6yz)})$

Step 1.4: Insert parentheses around the entire radical

$7\sqrt{3yz} \cdot (4\sqrt{2^2 \cdot (6yz)})$

Step 2: Simplify the radical expression

$7\sqrt{3yz} \cdot (4(2\sqrt{6yz}))$

Step 3: Multiply the coefficients outside the radical

$7\sqrt{3yz} \cdot (8\sqrt{6yz})$

Step 4: Perform multiplication of the radical expressions

$7\sqrt{3yz}(8\sqrt{6yz})$

Step 4.1: Multiply the coefficients

$56\sqrt{3yz}\sqrt{6yz}$

Step 4.2: Apply the product rule for radicals

$56\sqrt{6yz(3yz)}$

Step 4.3: Multiply the numbers under the radical

$56\sqrt{18yz(yz)}$

Step 4.4: Apply exponent rules to $y$

$56\sqrt{18(y^1y)z^2}$

Step 4.5: Apply exponent rules to $y$

$56\sqrt{18(y^1y^1)z^2}$

Step 4.6: Combine the exponents for $y$

$56\sqrt{18y^{1+1}z^2}$

Step 4.7: Add the exponents for $y$

$56\sqrt{18y^2z^2}$

Step 4.8: Apply exponent rules to $z$

$56\sqrt{18y^2(z^1z)}$

Step 4.9: Apply exponent rules to $z$

$56\sqrt{18y^2(z^1z^1)}$

Step 4.10: Combine the exponents for $z$

$56\sqrt{18y^2z^{1+1}}$

Step 4.11: Add the exponents for $z$

$56\sqrt{18y^2z^2}$

Step 5: Rewrite the expression under the radical

Express $18y^2z^2$ as $(3yz)^2 \cdot 2$.

Step 5.1: Factor out $9$ from $18$

$56\sqrt{9(2)y^2z^2}$

Step 5.2: Represent $9$ as $3^2$

$56\sqrt{3^2 \cdot 2y^2z^2}$

Step 5.3: Rearrange the terms

$56\sqrt{3^2y^2z^2 \cdot 2}$

Step 5.4: Rewrite $3^2y^2z^2$ as $(3yz)^2$

$56\sqrt{(3yz)^2 \cdot 2}$

Step 6: Simplify the radical

$56(3yz\sqrt{2})$

Step 7: Multiply the coefficients

$168yz\sqrt{2}$

Knowledge Notes:

To simplify the expression $7\sqrt{3yz} \cdot 4\sqrt{24yz}$, we follow these steps:

1. Decomposition of Numbers: Break down composite numbers into prime factors or identify square factors that can be easily taken out of the radical.

2. Simplification of Radicals: Use the property that $\sqrt{a^2} = a$ to simplify the expression under the radical.

3. Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$ allows us to combine radicals under a single radical sign.

4. Exponent Rules: $a^m \cdot a^n = a^{m+n}$ is used to combine like terms with exponents.

5. Multiplication of Coefficients: Multiply numbers outside the radical to consolidate the expression.

6. Final Simplification: After pulling out terms from under the radical and multiplying coefficients, we arrive at the simplest form of the expression.