Problem 44 - Elementary Algebra

Question

Simplify \(\sqrt{(x+3)^{4}(x-2)^{6}}\).

Step-by-Step Solution

Answer

Answer: \((x+3)^{2}(x-2)^{3}\)

Step 1: Identify the square root property

Recall the square root property, which states that \(\sqrt{a^2} = a\). This can also be written as \((a^2)^\frac{1}{2} = a\).

Step 2: Apply the square root property to the entire expression

First, to simplify the expression, rewrite the square root as an exponent of the entire expression: \(\sqrt{(x+3)^{4}(x-2)^{6}} = ((x+3)^{4}(x-2)^{6})^\frac{1}{2}\)

Step 3: Use exponent properties to simplify

Now, applying exponent properties, we can simplify the expression further: $$((x+3)^{4}(x-2)^{6})^\frac{1}{2} = (x+3)^{4\cdot\frac{1}{2}}(x-2)^{6\cdot\frac{1}{2}}$$

Step 4: Simplify exponents

Simplify the exponents, we get: $$(x+3)^{4\cdot\frac{1}{2}}(x-2)^{6\cdot\frac{1}{2}} = (x+3)^{2}(x-2)^{3}$$

Step 5: Write the final simplified expression

Thus, the simplified expression is: \((x+3)^{2}(x-2)^{3}\).

Key Concepts

Square Root Property

The square root property is a fundamental tool in algebra. It helps us simplify expressions by converting them into a more manageable form. Essentially, if we have a term like \((a^2)\), taking the square root gives us \(a\). This is because the square root and squaring are inverse operations. In our exercise, we start with an expression under a square root:

• \(\sqrt{(x+3)^{4}(x-2)^{6}}\)

To simplify this, we recognize that the entire expression can be raised to the power of \(\frac{1}{2}\), due to the square root property. This step forms the basis for further simplification using other algebraic rules.

Exponent Properties

Understanding and using exponent properties is key to successfully simplifying expressions. Exponents tell us how many times to multiply a number by itself, and we have several rules to manipulate them:

• Product of Powers Rule: When multiplying two powers that have the same base, add their exponents.
• Power of a Power Rule: To raise a power to another power, multiply the exponents.
• Power of a Product Rule: Distribute an exponent to each term within a product.

In our problem, we apply the Power of a Power Rule. We simplify the inside of the square root by raising the expression to the power of \(\frac{1}{2}\). This looks like multiplying each of the internal exponents by \(\frac{1}{2}\):

• \((x+3)^{4\cdot\frac{1}{2}}\) becomes \((x+3)^2\)
• \((x-2)^{6\cdot\frac{1}{2}}\) becomes \((x-2)^3\)

With these properties, we methodically reduce the complexity of the expression.

Algebraic Simplification

Algebraic simplification is the process of transforming an expression into its simplest or most elegant form. This often involves several steps of manipulating exponents, terms, and constants to reduce clutter and reveal the core structure of an equation. For our expression, we've already used the square root and exponent properties. The next step is simply recognizing the result:

• From \(((x+3)^2)\) and \((x-2)^3\), we finalize the expression as \((x+3)^2(x-2)^3\).

Simplifying isn't just about making numbers smaller. It's about revealing the most direct expression of mathematical relationships. This could mean reduced terms, factoring expressions, or—like in our case—combining properties like square roots and exponents to streamline calculations.