Problem 52 - Intermediate Algebra

Question

Simplify. See Examples 3 and 4 $$ \sqrt[3]{40 y^{10}} $$

Step-by-Step Solution

Answer

The simplified form is \(2y^3 \cdot \sqrt[3]{5y}\).

Step 1: Break Down the Expression

We start with the expression \( \sqrt[3]{40 y^{10}} \). Recognize 40 as \( 2^3 \times 5 \) and the exponent 10 of \( y^{10} \) can be rewritten as \( y^{3\times3 +1} = (y^3)^3 \times y \,^{1} \). Thus, the expression becomes \( \sqrt[3]{2^3 \times 5 \times (y^3)^3 \times y} \).

Step 2: Apply the Cube Root to Each Part

Apply the cube root to each part separately. The cube root of \(2^3\) is \(2\), the cube root of \(5\) remains \( \sqrt[3]{5} \), the cube root of \((y^3)^3\) simplifies to \(y^3\), and the cube root of \(y\) is \(\sqrt[3]{y}\).

Step 3: Combine Simplified Parts

Bring together all the simplified parts. Hence, the expression becomes \(2 \times y^3 \times \sqrt[3]{5} \times \sqrt[3]{y}\), which simplifies further to \(2y^3 \cdot \sqrt[3]{5y}\).

Key Concepts

Simplifying Expressions

When dealing with mathematical expressions, simplifying means breaking them down into their simplest form. This often involves figuring out common bases, common factors, or ways to combine terms so they appear cleaner and more manageable. It's like cleaning up your room: the individual parts might still all be there, but everything looks organized.
Begin by looking for terms or factors that can be easily combined. Consider their numerical coefficients and variables separately, as they might require different approaches. In this process:

• Identify any mathematical properties that can simplify calculations, for example, the distributive property or basic properties of roots and powers.
• Look for common factors or terms that might cancel out or combine with others.
• Keep an eye out for square roots, cube roots, or other radicals that might simplify.

Simplifying expressions makes them easier to work with, especially in equations or when you need to implement operations further along in solving a problem.

Exponents

Exponents are a powerful way to express repeated multiplication succinctly. For instance, saying 3 times 3 times 3 is the same as saying \(3^3\). This not only saves space but also makes calculations more straightforward, especially when dealing with large numbers or with many repetitions.
When working with exponents, a few simple rules can make calculations much easier:

• Product of Powers Rule: Add the exponents when multiplying with the same base, such as \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: Multiply the exponents when taking a power of a power, as in \((a^m)^n = a^{m \cdot n}\).
• Negative Exponents: Represent reciprocal values, like \(a^{-n} = \frac{1}{a^n}\).

Understanding these rules is crucial when simplifying expressions or altering the forms of numbers in any algebraic context. For cube roots, specifically, you're tackling an exponent of \(\frac{1}{3}\), rearranging terms to optimize the expression using these rules.

Factorization

Factorization involves breaking a composite number or expression down into its constituent parts, or factors, which when multiplied together give the original number or expression. It's a little like taking things apart to see what they're made of, which is crucial for simplification.
In algebra, factorization is often used to reduce expressions or solve equations. You might:

• Break down numbers into their prime factors, such as breaking 40 down into \(2^3 \times 5\).
• Factor out common elements in polynomials or algebraic expressions to simplify them.
• Use special factorization formulas like the difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).

By breaking down expressions into more manageable chunks, you're often able to simplify calculations dramatically. In the case of cube roots, identifying factors within the radicand can streamline the simplification process.