In Exercises 91–100, simplify using properties of exponents. (25x⁴y⁶)^1/2

Verified step by step guidance

1. Start by recognizing that the expression (25x^4y^6)^(1/2) involves applying the property of exponents: (a^m)^n = a^(m*n). This property allows us to simplify the expression by distributing the exponent (1/2) to each factor inside the parentheses.
2. Break the expression into its components: 25, x^4, and y^6. Apply the exponent (1/2) to each of these factors individually: (25^(1/2)), (x^4)^(1/2), and (y^6)^(1/2).
3. Simplify each term: For 25^(1/2), recall that the square root of 25 is 5, so 25^(1/2) = 5. For (x^4)^(1/2), use the property of exponents to multiply the exponents: 4 * (1/2) = 2, so (x^4)^(1/2) = x^2. Similarly, for (y^6)^(1/2), multiply the exponents: 6 * (1/2) = 3, so (y^6)^(1/2) = y^3.
4. Combine the simplified terms: Multiply the results from the previous step together: 5 * x^2 * y^3.
5. Write the final simplified expression: The simplified form of (25x^4y^6)^(1/2) is 5x^2y^3.

Key Concepts

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^n). Understanding these properties is essential for simplifying expressions with exponents.

Square Roots and Exponents

The square root of a number can be expressed as an exponent of one-half. For example, √a = a^(1/2). This relationship allows us to rewrite expressions involving square roots in terms of exponents, facilitating simplification. Recognizing this connection is crucial when simplifying expressions like (25x^4y^6)^(1/2).

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form by combining like terms and applying mathematical operations. This process often includes factoring, distributing, and using properties of exponents. Mastery of simplification techniques is vital for solving equations and understanding more complex algebraic concepts.