In Exercises 65–74, simplify each radical expression and then rationalize the denominator.5m⁴n⁶√ -------------15m³n⁴

Verified step by step guidance

1. Step 1: Simplify the expression under the square root. The expression is \( \frac{5m^4n^6}{15m^3n^4} \). Start by simplifying the fraction by dividing the coefficients and subtracting the exponents of like bases.
2. Step 2: Divide the coefficients: \( \frac{5}{15} = \frac{1}{3} \).
3. Step 3: Simplify the exponents of \( m \): \( m^{4-3} = m^1 \) or simply \( m \).
4. Step 4: Simplify the exponents of \( n \): \( n^{6-4} = n^2 \).
5. Step 5: The simplified expression under the square root is \( \frac{mn^2}{3} \). Now, rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{3} \) to eliminate the square root in the denominator.

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are often represented with the radical symbol (√). Simplifying radical expressions requires identifying perfect squares or cubes within the radicand (the expression under the root) to reduce the expression to its simplest form. Understanding how to manipulate these expressions is crucial for solving problems involving radicals.

Rationalizing the Denominator

Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable radical that will result in a rational number in the denominator. This technique is important in algebra to simplify expressions and make them easier to work with.

Exponents and Their Properties

Exponents represent repeated multiplication of a base number and have specific rules that govern their manipulation, such as the product of powers, quotient of powers, and power of a power. In the context of simplifying radical expressions, understanding how to convert between radical and exponent forms is essential, as radicals can be expressed as fractional exponents, facilitating simplification and rationalization.