Problem 48 - Elementary Algebra

Question

Find each of the following products. $$ \sqrt{x^{8}} \sqrt{x^{5}} $$

Step-by-Step Solution

Answer

Answer: The final simplified expression is \(x^\frac{13}{2}\).

Step 1: Simplify square roots

First, we need to simplify the square roots individually. Since both expressions have the same base (x), we can use the rule that \(\sqrt{x^a} = x^\frac{a}{2}\). Applying this rule to each expression, we get: $$ \sqrt{x^8} = x^\frac{8}{2} = x^4 $$ and $$ \sqrt{x^5} = x^\frac{5}{2} $$

Step 2: Multiply the simplified expressions

Now that we have simplified each expression, we can multiply them together. When multiplying expressions with the same base, we add the exponents. So, using that rule, we multiply \(x^4\) by \(x^\frac{5}{2}\): $$ x^4 \cdot x^\frac{5}{2} = x^{4 + \frac{5}{2}} $$

Step 3: Simplify the equation

To simplify the expression further, we need to find a common denominator for the exponents to add them together. In this case, we can rewrite the exponent of 4 as \(\frac{8}{2}\): $$ x^{4 + \frac{5}{2}} = x^{\frac{8}{2} + \frac{5}{2}} = x^\frac{13}{2} $$

Step 4: Final Answer

Thus, the final simplified expression after multiplying the square roots is: $$ \sqrt{x^{8}} \sqrt{x^{5}} = x^\frac{13}{2} $$

Key Concepts

Understanding Exponents

Exponents are an essential part of algebra that indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(x^8\), the base is \(x\) and the exponent is \(8\). This means \(x\) is to be multiplied by itself 8 times. Exponents follow specific rules to make calculations more straightforward:

• Multiplying with same bases: When you multiply expressions like \(x^a\) and \(x^b\), you add the exponents: \(x^{a+b}\).
• Dividing with same bases: Dividing expressions like \(x^a\) by \(x^b\) involves subtracting the exponents: \(x^{a-b}\).
• Power raised to power: If you have an exponent raised to another exponent, say \((x^a)^b\), you multiply the exponents: \(x^{a\cdot b}\).

Exponents are a powerful tool for handling large numbers and algebraic expressions, allowing for simplified expression management and calculations.

Exploring Square Roots

Square roots are another fundamental concept in algebra used to find a number that, when multiplied by itself, gives the original number. The square root of \(x^2\) is \(x\), since \(x\times x = x^2\). For square roots involving exponents like \( \sqrt{x^8} \), it is useful to remember the rule \( \sqrt{x^a} = x^{\frac{a}{2}} \). This rule helps in simplifying square roots of algebraic expressions:

• Simplifying \( \sqrt{x^8} \): Here, the expression simplifies to \( x^{\frac{8}{2}} = x^4 \).
• Simplifying \( \sqrt{x^5} \): This is simplified to \( x^{\frac{5}{2}} \).

Understanding how to handle square roots can drastically reduce the complexity of algebraic problems and is especially useful when the base remains consistent across multiple terms.

The Art of Simplifying Expressions

Simplifying expressions is a crucial skill in algebra that involves rewriting a complex expression in its simplest form without changing its value. The goal is to make the expression easier to understand or further calculate. In the exercise provided, simplifying involves using exponent rules effectively:

• First, simplify internal components like square roots to manageable exponents using rules such as \( \sqrt{x^a} = x^{\frac{a}{2}} \).
• Next, combine like terms by adding or subtracting their exponents when they share the same base. For example, \( x^4 \cdot x^{\frac{5}{2}} = x^{4 + \frac{5}{2}} \).
• Ensure the final expression presents exponents in their simplest form, such as converting whole numbers to fractions to manage addition, like \( x^{4 + \frac{5}{2}} = x^{\frac{13}{2}} \).

Mastering these techniques can streamline problem-solving and significantly enhance your ability to tackle complex algebraic expressions.