Problem 39 - Algebra 2

Question

Simplify each expression. Rationalize all denominators. Assume that all variables are positive. $$ \sqrt{x^{5} y^{5}} \cdot 3 \sqrt{2 x^{7} y^{6}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression with all denominators rationalized is \(3x^{6} \cdot y^{5} \cdot \sqrt{y}\).

Step 1: Group terms with the same bases

Group same bases under square root together. We have \(x\) and \(y\) as bases in both expressions. So we can write the expression as \(\sqrt{x^{5} \cdot 2x^{7}} \cdot \sqrt{y^{5} \cdot y^{6}} \cdot 3\).

Step 2: Simplify exponents and square roots

Apply the law of exponents for multiplication, then simplify square roots. This gives us \(\sqrt{x^{12}} \cdot \sqrt{y^{11}} \cdot 3\). Next, apply the square root to the exponents, resulting in \(x^{6} \cdot y^{5.5} \cdot 3\).

Step 3: Rationalize the denominator

The expression \(y^{5.5}\) is not in a rational form. Multiply by the necessary term to rationalize this expression, which is \(\sqrt{y}\). This gives us \(3x^{6} \cdot y^{5} \cdot \sqrt{y}\).

Step 4: Simplify the final expression

The result from step 3 is the simplified form of the original expression with all denominators rationalized.

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing them down to their simplest form. It's like cleaning up a messy room, making everything neat and tidy.

When simplifying, you combine like terms and eliminate unnecessary elements. For example, if you have terms with the same base under a square root, like \( \sqrt{x^5 \cdot 2x^7} \), you can simplify them by using laws of exponents to consolidate terms prior to further simplification.

• Group terms with the same base together.
• Use arithmetic operations to simplify them as far as you can.
• Simplify any roots or fractional exponents next.

In our exercise, after combining bases under the square root, we end up with something more manageable, \( \sqrt{x^{12}}, \sqrt{y^{11}} \), which can be further simplified by applying the square roots.

Laws of Exponents

The laws of exponents provide rules for simplifying expressions with powers involving the same base. This is extremely helpful when you have complex expressions with multiple exponents.

There are several rules, but some of the key ones applied in simplifying our expression include:

• Product of Powers Rule: When multiplying like bases, you add their exponents. For example, \(x^a \cdot x^b = x^{a+b}\).
• Power of a Power Rule: When raising a power to another power, you multiply the exponents. For instance, \((x^a)^b = x^{a \cdot b}\).

In the original problem, we used the Product of Powers Rule to simplify \(\sqrt{x^{5} \cdot 2x^{7}}\) into \(\sqrt{x^{12}}\) and \(\sqrt{y^{5} \cdot y^{6}}\) into \(\sqrt{y^{11}}\). This makes further steps of simplification straightforward.

Rational Expressions

A rational expression is essentially a fraction where the numerator and the denominator are both polynomials. When dealing with rational expressions, rationalizing the denominator means ensuring that no roots or radicals remain in the denominator.

In many advanced algebra problems, it's essential to express the expression such that all terms in the denominator are rationalized—this means they should not contain any irrational numbers such as square roots.

• Identify what is irrational in the denominator.
• Multiply the numerator and the denominator by a term that will rationalize it.
• Simplify to ensure the final expression is in its simplest form.

In our example, the step of multiplying by \(\sqrt{y}\) ensures that any irrational numbers are moved out of the denominator, thus giving a simplified form \(3x^6 \cdot y^5 \cdot \sqrt{y}\). This makes computations cleaner and often required in both theoretical and practical problems.