Problem 44 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{x^{3}} \sqrt{x^{7}} $$

Step-by-Step Solution

Answer

Question: Simplify the given expression: $$\sqrt{x^{3}} \cdot \sqrt{x^{7}}$$ Answer: The simplified expression is $$x^5$$.

Step 1: Multiply the square roots together

In this step, we will multiply both given square roots together: $$ \sqrt{x^{3}} \cdot \sqrt{x^{7}} $$

Step 2: Use the property of exponents to combine the square roots

Since both expressions have the same base (x), we can combine them using the following property: $$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$ Applying this property in our expression, we obtain: $$ \sqrt{x^{3} \cdot x^{7}} $$

Step 3: Simplify the multiplication of the exponents

To multiply the bases with their exponents, we use the rule $$x^a \cdot x^b = x^{a+b}$$, so in our case, we will add the exponents: $$ \sqrt{x^{3+7}} $$ Which simplifies to: $$ \sqrt{x^{10}} $$

Step 4: Simplify the square root expression

Now, we need to simplify the expression further. Since the exponent is even, we can take half of the exponent as the new exponent: $$ \sqrt{x^{10}} = x^{10/2} $$ So, the final answer is: $$ x^5 $$

Key Concepts

Properties of Exponents

To understand the multiplication of square roots involving variables, it's crucial to grasp the properties of exponents. Exponents inform how many times a base is used as a factor. Let's highlight a pivotal exponent property essential for our problem:

• The Product of Powers Property: This states that when you multiply two expressions with the same base, you add the exponents: \( a^m \times a^n = a^{m+n} \). This property helps simplify expressions easily by turning repeated multiplication into addition of exponents.

For instance, for \( x^3 \times x^7 \), since the base "x" is the same, add up the exponents: \( 3 + 7 = 10 \), resulting in \( x^{10} \). This property becomes invaluable when dealing with square roots of expressions with the same base because it allows us to simplify without explicitly calculating each power.

Simplifying Square Roots

Once you have a firm grasp of exponents, the next step is simplifying square roots. Square roots involve finding a number which, when multiplied by itself, delivers the original number under the root. We can apply certain rules to simplify square roots:

• The square root of a product can be split into the product of separate roots: \( \sqrt{a \times b} = \sqrt{a} \cdot \sqrt{b} \). This enables us to handle each part separately before recombining them.
• If an exponent is even under a square root, it can be divided by 2 for simplification: \( \sqrt{x^{2m}} = x^m \).

When applying these rules to \( \sqrt{x^{10}} \), because the exponent 10 is even, the expression simplifies directly: \( \sqrt{x^{10}} = x^{5} \). This involves halving the exponent because we are essentially reversing multiplication that happens under a square. Such simplification steps are crucial, especially when faced with more complex algebraic expressions under square roots.

Algebraic Simplification

Algebraic simplification combines understanding properties of exponents and rules for square roots to distill complex expressions into simpler, more workable forms. Let's break down the process: - **Combine like terms appropriately**: Identify expressions with the same base and apply exponent rules to consolidate them, as seen in multiplying \( x^3 \) and \( x^7 \) to get \( x^{10} \). - **Apply simplification rules**: Use the property of square roots to manage each variable or numerical component under the root. With \( \sqrt{x^{10}} \), since 10 is even, divide the exponent by 2. - **Check for final neatness**: Always simplify further if possible, to express the expression in its simplest form, which, in this case, results in \( x^5 \). Each step aims to progressively ease the complexity, making difficult algebraic expressions much simpler and more straightforward to evaluate. Approaching problems step by step ensures clarity and reduces the chance of errors.