Problem 46 - Algebra 2

Question

Simplify each expression. $$ \sqrt[8]{25 x^{4} y^{4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5^{1/4} x^{1/2} y^{1/2} \).

Step 1: Understand the Root and the Expression

We need to simplify the expression \( \sqrt[8]{25 x^{4} y^{4}} \). This expression contains an 8th root of a product, which means we need to find another expression that is equivalent to this root.

Step 2: Separate the Radicand into Factors

The expression \( 25 x^{4} y^{4} \) inside the root can be factored into separate parts: \( 25 = 5^2 \), \( x^4 \), and \( y^4 \). Therefore, \( 25 x^{4} y^{4} = (5^2)(x^4)(y^4) \).

Step 3: Apply the Power of a Product Rule

The root of the product rule states that \( \sqrt[n]{a \, b} = \sqrt[n]{a} \, \sqrt[n]{b} \). So, we have \( \sqrt[8]{25 x^{4} y^{4}} = \sqrt[8]{5^2} \cdot \sqrt[8]{x^4} \cdot \sqrt[8]{y^4} \).

Step 4: Simplify Each Component

Simplify each root separately: - \( \sqrt[8]{5^2} \) stays as it is because \( 5^2 \) does not apply to the 8th root directly. - \( \sqrt[8]{x^4} = x^{4/8} = x^{1/2} \), because \( 4/8 = 1/2 \). - Similarly, \( \sqrt[8]{y^4} = y^{1/2} \).

Step 5: Combine the Simplified Components

Combine the components to get \( \sqrt[8]{5^2} \cdot x^{1/2} \cdot y^{1/2} \). This simplifies to \( 5^{1/4} \cdot x^{1/2} \cdot y^{1/2} \) because \( (5^2)^{1/8} = 5^{2/8} = 5^{1/4} \).Thus, the simplified expression is \( 5^{1/4} x^{1/2} y^{1/2} \).

Key Concepts

Understanding nth Roots

Nth roots are similar to square roots but apply to any integer 'n'. They ask "what number do we multiply by itself 'n' times to get the original number?".
For example, the cube root entails finding a number that multiplies by itself three times to reach the original figure.
In the expression \( \sqrt[8]{25 x^{4} y^{4}} \), the 8th root means we are looking for something that will equate to the original input when raised to the power of 8.
That's why each factor inside the root is separately simplified using roots.

• A root can break down complex expressions and identify simpler equivalent expressions.
• The same rules apply from simple square roots; however, nth roots can handle higher powers.

When working with nth roots, your outcome may still contain an expression in root form, especially if the powers do not evenly divide by 'n'.

The Role of Exponents

Exponents represent repeated multiplication of a base number. For instance, \( x^4 \) means \( x \) multiplied by itself 4 times.
Exponent rules help in handling roots and simplifying expressions, especially when working with prime factors.

• When taking nth roots, you might use the property \( a^{m/n} = (a^m)^{1/n} = \sqrt[n]{a^m} \).
• The power of a product rule, \( (ab)^n = a^n \, b^n \), allows the separation of exponents within a root.

By converting nth roots into fractional exponents, you often simplify the computation, as seen with \( \sqrt[8]{x^4} = x^{1/2} \).
Fractional exponents can make working with and understanding roots more intuitive.

Simplifying by Factorization

Factorization involves breaking down numbers or expressions into their simplest building blocks, which can include primes or polynomial factors.
Factorizing is helpful for simplifying expressions under a root by reorganizing them into parts that the root can easily manage.

• For example, \( 25 = 5^2 \), and in the given expression, \( x^4 = (x^2)^2 \) is already factored accordingly.
• Once in factored form, apply rules of exponents and roots more easily.

Using factorization eases the process of dividing exponents by the root's degree, enabling simplification, like reducing \( (5^2)^{1/8} \) to \( 5^{1/4} \).
Thus, factorization is a critical step in reducing the complexity of radical expressions.