Problem 305 - Elementary Algebra

Question

In the following exercises, simplify. $$ \left(\sqrt{35 y^{3}}\right)\left(\sqrt{7 y^{3}}\right) $$

Step-by-Step Solution

Answer

7y^3 \sqrt{5}

Step 1: Understand the problem

We need to simplify the product of the square roots: \(\sqrt{35 y^{3}} \times \sqrt{7 y^{3}}\).

Step 2: Use the property of square roots

Recall that \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). Applying this property, \( \sqrt{35 y^{3}} \times \sqrt{7 y^{3}} = \sqrt{(35 y^{3}) \times (7 y^{3})} \).

Step 3: Multiply the terms inside the square root

Multiply the constants and the variables separately: \( (35 y^{3}) \times (7 y^{3}) = 245 y^{6} \).

Step 4: Simplify the square root

Simplify \( \sqrt{245 y^{6}} \). First, factor the term inside the square root: \( 245 = 5 \times 7 \times 7 \). Therefore, \( \sqrt{245 y^{6}} = \sqrt{5 \times 7^2 \times y^6} \).

Step 5: Extract the squares

Extract the factors that are perfect squares: \( \sqrt{5 \times 7^2 \times y^6} = 7y^3 \sqrt{5} \).

Step 6: Write the final simplified form

The simplified form of the given expression is \( 7y^3 \sqrt{5} \).

Key Concepts

Understanding Square Roots

Square roots are numbers that, when multiplied by themselves, give the original number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. The symbol for the square root is \( \sqrt{} \). It's important to remember that not every number has a perfect square root.

In expressions like \(\sqrt{35 y^{3}}\), we are taking the square root of both a constant (35) and a variable with an exponent (\(y^3\)). To understand this better, you can think of breaking down the constant into its prime factors and seeing if any of those factors form pairs, which makes them perfect squares and can be simplified outside the square root.

This basic knowledge of square roots will help you understand how to handle more complex expressions involving roots.

Product of Square Roots

When you multiply two square roots, you can use an important property: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \). This property allows you to combine the two roots into one. For example, in the expression \(\sqrt{35 y^{3}} \times \sqrt{7 y^{3}}\), you can merge it into one square root: \(\sqrt{(35 y^{3}) \times (7 y^{3})}\).

First, multiply the constants: 35 times 7 equals 245. Then, for the variables, you add their exponents as you multiply: \(y^3 \times y^3 = y^{6}\). So, inside the square root now, you have \(245 y^{6}\).

This key property of square roots simplifies the whole process, making it straightforward to combine and then simplify further.

Simplification Steps

Simplifying square root expressions involves several clear steps. Let's revisit the example: \(\sqrt{35 y^{3}} \times \sqrt{7 y^{3}}\).

• Step 1: Understand and rewrite the expression using properties of square roots: \(\sqrt{35 y^{3}} \times \sqrt{7 y^{3}} = \sqrt{(35 y^{3}) \times (7 y^{3})}\).


• Step 2: Multiply inside the square root: \(\sqrt{(35 \times 7) \times (y^{3} \times y^{3})} = \sqrt{245 y^{6}}\).


• Step 3: Factorize the number 245 to identify perfect squares: \(245 = 5 \times 7 \times 7\). Now the expression is \(\sqrt{5 \times 7^2 \times y^6}\).


• Step 4: Extract the perfect squares from the square root: \(\sqrt{7^2} = 7\) and \(\sqrt{y^6} = y^3\). This gives us: \(7y^3 \sqrt{5}\).

So, \(\sqrt{35 y^{3}} \times \sqrt{7 y^{3}}\) simplifies to \(7y^3 \sqrt{5}\).

Following these steps ensures you systematically simplify and don't miss any details.