Problem 61 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{25 a^{2} b^{20}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 5ab^{10} \).

Step 1: Understand the Square Root Function

The square root function is searching for a number that, when multiplied by itself, equals the given expression under the root. For example, the square root of 25 is 5 because 5 multiplied by 5 equals 25.

Step 2: Simplify the Number under the Root

Identify any perfect square factors under the radical. In this case, within the expression \( \sqrt{25 a^2 b^{20}} \), the number 25 can be simplified as it is a perfect square, so \( \sqrt{25} = 5 \).

Step 3: Simplify the Variables under the Root

Apply the square root to each variable separately:- \( a^2 \) is a perfect square: \( \sqrt{a^2} = a \).- \( b^{20} \) can be rewritten as \( (b^{10})^2 \), allowing us to write: \( \sqrt{b^{20}} = b^{10} \).

Step 4: Combine Simplified Elements

Combine elements from Steps 2 and 3. The simplified expression is: \[ \sqrt{25 a^{2} b^{20}} = 5ab^{10} \].

Key Concepts

Understanding Square Roots

The square root is a fascinating mathematical operation. It asks "what number, multiplied by itself, gives this value?" For the number 25, its square root is 5 because multiplying 5 by itself gives 25. Think of it as the opposite of squaring a number. This operation is most often seen as the symbol \( \sqrt{} \).

Here are some things to remember about square roots:

• If the number inside the square root (known as the radicand) is a perfect square, the square root simplifies to a whole number.
• When working with variables, you apply the square root to each part separately.
• Understanding square roots helps simplify expressions in algebra.

Square roots are important in mathematics because they are used in computations involving areas and solutions of equations. Mastering square roots simplifies many algebra problems.

The Role of Perfect Squares

Perfect squares are the numbers that gain importance when dealing with square roots. A perfect square is simply a number multiplied by itself. Examples include 1, 4, 9, 16, 25, etc. These numbers already have a nice neat square root.

When simplifying expressions under a radical, such as \( \sqrt{25} \), recognizing 25 as \( 5^2 \) tells us that its square root is simply 5. For variables in radical expressions, it’s the same principle. In our example, \( a^2 \) is a perfect square, so \( \sqrt{a^2} \) simplifies to \( a \).

The key benefits of identifying perfect squares include:

• They help eliminate the radical in the expression entirely.
• They simplify your calculations by reducing complex terms into simple numerical forms.

Recognizing perfect squares quickly is a skill that aids greatly in algebra and further math studies.

Dealing with Variable Exponents

Variable exponents can seem tricky at first, but understanding them simplifies many algebraic expressions. When you see a variable with an exponent within a radical, remember that you're looking for manageable ways to reduce it.

Take \( b^{20} \), for example. Since we're taking the square root, we're looking for something squared that, when squared, gives us \( b^{20} \). This can be written as \( (b^{10})^2 \). Thus, \( \sqrt{b^{20}} = b^{10} \).

Here are some quick tips for working with variable exponents:

• Split the exponent into two parts that are easy to manage, like squared terms.
• Ensure every part of the exponent has been simplified according to the root you're taking.
• Always simplify the expression fully by individually handling each variable.

Understanding exponents fully allows for both clarity and efficiency when simplifying expressions.