Problem 91 - College Algebra

Question

In Exercises \(85-94,\) simplify using properties of exponents. $$\left(25 x^{4} y^{6}\right)^{1 / 2}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( (25 x^{4} y^{6})^{1 / 2} \) is \(5x^{2}y^{3}\).

Step 1: Square Root of the Constant

Take the square root of the constant 25 which simplifies to 5 since \( \sqrt{25} = 5 \).

Step 2: Apply Power Rule for Variable with Exponent x

Remember the power rule: \( (a^{m})^{n} = a^{mn} \). In this case, apply this rule to \( x^4 \) giving us \( x^{4*1/2} = x^{2} \) since multiplication of 4 and 1/2 is 2.

Step 3: Apply Power Rule for Variable with Exponent y

Similarly, we apply the power rule to \( y^6 \) resulting in \( y^{6*1/2} = y^{3} \) since multiplication of 6 and 1/2 is 3.

Step 4: Combine the Results

Combining the results from Steps 1, 2, and 3, we get the solution as \( 5x^{2}y^{3} \). This is the simplified form of the original expression.

Key Concepts

Simplifying Expressions

Simplifying expressions in mathematics means to reduce a complex mathematical expression into its simplest form. This often involves combining like terms, applying properties of operations, and using properties of exponents, which can transform a bulky expression into something more manageable and easier to work with. For example, simplifying \( (25x^4y^6)^{1/2} \) involves taking each element within the parentheses—the constant, and the variables to their respective powers—and applying the square root to them individually.

In practice, this skill is essential as it makes the subsequent arithmetic operations easier and can reveal more information about the mathematical relationship within the expression. For students, mastering simplification can lead to a better understanding of algebra and other areas of mathematics where expressions become increasingly complex.

Power Rule of Exponents

The power rule of exponents is a foundational principle in algebra that simplifies the process of raising a power to another power. It states that when you take an exponent to another exponent, you multiply the exponents together. The general form of this rule is expressed as \( (a^m)^n = a^{mn} \) where \( a \) is the base and \( m \) and \( n \) are the exponents.

For instance, in simplifying \( x^4 \) raised to the power of \( 1/2 \) as shown in the exercise solution, we multiply the exponents \( 4 \) and \( 1/2 \) to get \( x^{2} \). This rule is incredibly powerful as it works for any real numbers and applies regardless of whether the exponents are positive, negative, whole numbers, or fractions.

Square Root Simplification

Square root simplification deals with the process of finding the number that, when multiplied by itself, gives the original number. The square root of a number is denoted by \( \sqrt{x} \) where \( x \) is the number. When we simplify square roots, we look for perfect squares that factor out of the number or expression under the root. In the exercise, the square root of 25 is taken, yielding 5, since \( 5 \times 5 = 25 \).

Moreover, when the square root is applied to variables with exponents under the radical symbol, it's equivalent to dividing the exponent by 2, since \( (x^m)^{1/2} = x^{m/2} \). This makes the square root one of the key tools in simplifying algebraic expressions with exponents, turning what may seem like a daunting problem into a much simpler form.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. When dealing with algebraic expressions, it's crucial to understand the rules and properties of arithmetic to manipulate them suitably. In our example, we worked with the expression \( (25x^4y^6)^{1/2} \) which is a combination of constants and variables raised to powers.

These types of expressions are the bedrock of algebra and provide a framework for describing relationships and solving problems involving unknown values. By using various properties, such as the distributive property, the associative property, and the power rule of exponents, students can unravel complex expressions to reveal a more comprehendible mathematical structure.