Question

Evaluate the expression when \(x=2, y=3\), and \(z=5$$\sqrt{3 x^{2} y z^{6}}\)

Short answer

The evaluated expression is 750.

Step-by-step solution

Substitute the Values

Substitute \(x = 2, y = 3\), and \(z = 5\) into the expression \(\sqrt{3 x^{2} y z^{6}}\). The expression becomes \(\sqrt{3 \cdot 2^{2} \cdot 3 \cdot 5^{6}}\).

Simplify the Expression

Next, calculate the values inside the square root. \(2^{2} = 4\), \(5^{6} = 15625\). Substitute these values back into the expression to get \(\sqrt{3 \cdot 4 \cdot 3 \cdot 15625}\).

Calculate the Final Value

Multiply the numbers to get \(\sqrt{3 \cdot 4 \cdot 3 \cdot 15625} = \sqrt{562500}\). Lastly, calculating the square root of this number gives the final value. The square root of 562500 is 750.

Key concepts

Substitution Method

The substitution method in algebra is a powerful tool used to evaluate expressions for specific values of variables. Understanding this method is essential because it allows us to find numerical results from algebraic expressions, making them more tangible.

Here's how it works: when you have an expression such as \(\sqrt{3x^2yz^6}\) and specific values for the variables \(x\), \(y\), and \(z\), the first step is to replace these variables with their given numbers. So, for the exercise at hand, set \(x = 2\), \(y = 3\), and \(z = 5\). After substituting these values, the expression transforms to \(\sqrt{3 \cdot 2^2 \cdot 3 \cdot 5^6}\).

This step is crucial as it forms the base for further calculation. Remember:

• Perform the substitutions carefully to prevent errors.
• Verify each substitution to check the accuracy.
• Write down the modified expression for clarity.

Simplification

Once you've substituted the values into the algebraic expression, the next step is simplification. Simplification involves breaking down complex parts of the expression into more understandable and manageable pieces, focusing on arithmetic operations such as exponentiation, multiplication, division, etc.

In our example, you begin by calculating the powers: \(2^2 = 4\) and \(5^6 = 15625\). The expression then becomes \(\sqrt{3 \cdot 4 \cdot 3 \cdot 15625}\).

Let's go through the possible simplification steps:

• Start with exponentiation, which simplifies the powers of numbers.
• Proceed to multiplication; calculate \(3 \cdot 4\) then multiply by \(3\), and finally by \(15625\).
• Ensure each simplification step is validated with accurate arithmetic, as errors here can compound and lead to incorrect results.

Simplification makes the expression easier to work with and lays the groundwork for evaluating the final value.

Square Roots

Understanding square roots in algebra is vital since they frequently appear in expressions that require evaluation. The square root function essentially "undoes" a square power, bringing us back to the root of a number. The symbol \(\sqrt{ }\) denotes the square root operation, and it asks which number squared will yield the original number.

After simplifying the expression to \(\sqrt{562500}\), the final step is to calculate the square root of 562500. If you're wondering how to do this effectively:

• First note that recognizing perfect squares helps determine the square roots faster.
• If a calculator is not available, use methods like prime factorization or approximation for manual calculation.
• Use a calculator for verification if possible. For 562500, it's confirmed that the square root is 750.

Square roots can sometimes seem tricky without practice, but regular use in algebra can help you become more comfortable with these calculations.