Problem 77 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \( 6y^3\sqrt{2y} \).

Step 1: Simplify Each Square Root Separately

Begin by simplifying each square root expression individually. Start with \( \sqrt{8y^7} \). This can be broken down as \( \sqrt{8} \times \sqrt{y^7} \). For \( \sqrt{8} \), simplify it as \( \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \). And for \( \sqrt{y^7} \), it becomes \( y^3\sqrt{y} \), because \( \sqrt{y^7} = y^{3.5} = y^3 \sqrt{y} \). Therefore, \( \sqrt{8y^7} = 2\sqrt{2} \times y^3 \sqrt{y} = 2y^3\sqrt{2y} \).

Step 2: Simplify the Second Square Root Expression

Now simplify \( \sqrt{32y^7} \). This can be expressed as \( \sqrt{32} \times \sqrt{y^7} \). For \( \sqrt{32} \), write it as \( \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \). The \( \sqrt{y^7} \) component remains \( y^3\sqrt{y} \). Hence, \( \sqrt{32y^7} = 4y^3\sqrt{2y} \).

Step 3: Combine the Simplified Expressions

Now add the two simplified terms from the previous steps: \( 2y^3\sqrt{2y} + 4y^3\sqrt{2y} \). Since both terms contain \( y^3 \sqrt{2y} \), you can combine them by adding the coefficients, which are 2 and 4. Thus, \( 2y^3\sqrt{2y} + 4y^3\sqrt{2y} = (2+4)y^3\sqrt{2y} = 6y^3\sqrt{2y} \).

Step 4: Write the Final Simplified Expression

The simplified expression for the original problem \( \sqrt{8y^7} + \sqrt{32y^7} \) is \( 6y^3\sqrt{2y} \). This is the sum of the simplified components from the previous steps.

Key Concepts

Square Roots

When we discuss square roots, we are trying to find a number that, when multiplied by itself, equals the original number. For example, the square root of 9 is 3, because 3 multiplied by 3 equals 9. We write this as \( \sqrt{9} = 3 \).

Understanding square roots is essential in simplifying expressions like \( \sqrt{8} \). To simplify \( \sqrt{8} \), we look for factors that form perfect squares. Here, \( 8 = 4 \times 2 \), and since \( \sqrt{4} = 2 \), this can be rewritten as \( 2\sqrt{2} \). This approach helps break down more complex square roots in algebraic expressions.

The square root simplification isn't just about numbers. It's also about algebraic terms such as \( \sqrt{y^7} \). Here, we use the rule \( \sqrt{y^n} = y^{n/2} \). So \( \sqrt{y^7} = y^{3.5} = y^3\sqrt{y} \). After simplifying each component, we can combine them back to form a neat expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables (like \( y \)), and operations (such as addition and multiplication). In simplifying algebraic expressions with square roots, we often look to factor them first.

Take \( \sqrt{8y^7} \) as an example from the exercise. We separate the numbers and the variable parts, simplifying each section individually. For the numeric part, \( \sqrt{8} \) becomes \( 2\sqrt{2} \), and for the variable part, \( \sqrt{y^7} \) turns into \( y^3\sqrt{y} \). The full expression simplifies to \( 2y^3\sqrt{2y} \).

As you continue learning, remember that working step-by-step helps ensure accuracy and understanding, especially when you have multiple terms in an expression like \( \sqrt{8y^7} + \sqrt{32y^7} \). Simplifying such algebraic expressions involves factoring and reorganizing terms in a way that makes them easier to combine.

Positive Real Numbers

Positive real numbers are all the numbers greater than zero. They include whole numbers, fractions, and irrational numbers like square roots. When simplifying expressions involving variables, assuming they're positive real numbers helps simplify the process since it means they don't result in undefined expressions like square roots of negative numbers.

In the exercise, we're reminded that the variables, like \( y \), are positive, allowing us to comfortably apply rules of exponents and roots without worrying about negative outcomes or extraneous solutions.

Working with positive real numbers in algebraic expressions ensures that every step follows the standard mathematical principles, helping prevent errors. With this, the expression \( \sqrt{8y^7} + \sqrt{32y^7} \) remains valid and simplifies smoothly to \( 6y^3\sqrt{2y} \). Understanding the properties of positive real numbers can enhance your ability to manipulate square roots and exponential expressions efficiently.