Problem 77 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{8 y^{7}}+\sqrt{32 y^{7}}-\sqrt{2 y^{7}} $$

Step-by-Step Solution

Answer

\( 5y^3 \sqrt{2y} \)

Step 1: Simplify Each Square Root Term

Start by simplifying each term under the square root by factoring out perfect squares. For \( \sqrt{8y^7} \), write it as \( \sqrt{4 \cdot 2 y^6 \cdot y} \) which simplifies to \( 2y^3 \sqrt{2y} \). For \( \sqrt{32y^7} \), write it as \( \sqrt{16 \cdot 2 y^6 \cdot y} \) which simplifies to \( 4y^3 \sqrt{2y} \). The term \( \sqrt{2y^7} \) can be expressed as \( \sqrt{2 \cdot y^6 \cdot y} \), simplifying to \( y^3 \sqrt{2y} \).

Step 2: Combine Like Terms

Once all terms are simplified, you have \( 2y^3 \sqrt{2y} + 4y^3 \sqrt{2y} - y^3 \sqrt{2y} \). Since they all share \( y^3 \sqrt{2y} \) as a common factor, you can combine them by adding and subtracting their coefficients: \( (2 + 4 - 1) y^3 \sqrt{2y} = 5y^3 \sqrt{2y} \).

Key Concepts

Understanding Square Roots

When you see a square root, it represents a number that, when multiplied by itself, gives you the value under the radical sign. A square root helps simplify expressions where you're dealing with squares. In the expression \( \sqrt{8y^7} \), the goal is to simplify the term by finding perfect square factors.

• A perfect square is a number that has an integer as its square root, like 4, 9, or 16.
• In \( \sqrt{8y^7} \), factor 8 into \( 4 \times 2 \) and treat \( y^7 \) as \( y^6 \times y \).
• This allows you to express it as \( \sqrt{4 \cdot 2 \cdot y^6 \cdot y} \), simplifying to \( 2y^3 \sqrt{2y} \).

Breaking down square roots in this manner reveals simpler forms and makes it easier to combine them with other similar terms.

Mastering Factorization

Factorization is like unlocking a hidden structure within numbers and expressions. It's a process of breaking down expressions into products of factors. For simplifying our expression, we factor perfect squares out of the square roots, making them easier to handle.

• Look for factors that are perfect squares and variables that can be written as powers of exponents.
• For \( \sqrt{32y^7} \), factor it as \( \sqrt{16 \times 2 \times y^6 \times y} \), resulting in \( 4y^3 \sqrt{2y} \).
• This process not only simplifies each root but also reveals common factors that can be combined later.

Learning to factor effectively enables you to streamline expressions and possibly resolve even more complex problems.

Combining Like Terms

Combining like terms is the technique of merging similar expression parts to simplify them. In the given exercise, after every square root is simplified, you notice that they all contain the same factor, \( y^3 \sqrt{2y} \).

• Each term, like \( 2y^3 \sqrt{2y} \) and \( 4y^3 \sqrt{2y} \), has the same "like" factor.
• To combine them, focus on the numerical coefficients of these terms.
• Add or subtract these coefficients: \( (2 + 4 - 1) y^3 \sqrt{2y} \) simplifies to \( 5y^3 \sqrt{2y} \).

Combining like terms helps consolidate your expression into a more manageable form. This step is vital in both algebraic simplifications and in making complex problems more approachable.