Problem 9 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \sqrt{9 b^{3}}-\sqrt{25 b^{3}}+\sqrt{49 b^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 5b^{3/2} \).

Step 1: Simplify Each Radical

Rewrite each term in the expression using the prime factorization of the coefficients and the variables inside the square roots. For instance: \[ \sqrt{9b^3} = \sqrt{(3^2)b^3} = 3b^{3/2} \] \[ \sqrt{25b^3} = \sqrt{(5^2)b^3} = 5b^{3/2} \] \[ \sqrt{49b^3} = \sqrt{(7^2)b^3} = 7b^{3/2} \]

Step 2: Combine Like Terms

Now that you have simplified each radical, notice that they all have a common term, which is \( b^{3/2} \). Thus, you can factor this out:\[ 3b^{3/2} - 5b^{3/2} + 7b^{3/2} = (3 - 5 + 7)b^{3/2} \] Simplify inside the parenthesis:\[ (3 - 5 + 7) = 5 \]

Step 3: Write the Final Solution

Substitute back the simplified result from Step 2: \[ 5b^{3/2} \] This expression represents the simplified form of the original problem.

Key Concepts

Prime Factorization

Prime factorization is the method of expressing a number as a product of its prime factors. This is particularly useful in simplifying expressions with radicals. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To find the prime factorization of a number, you keep dividing the number by the smallest prime numbers until you are left with 1.
For example, in the expression

• 9 can be expressed as \(3^2\),
• 25 as \(5^2\), and
• 49 as \(7^2\).

Using prime factorization helps when dealing with square roots or other radicals because it allows you to "break down" the inside of the root into manageable terms. You can then simplify square roots by taking out pairs of identical factors. In our exercise, this is how we simplified \( \sqrt{9b^3} \), \( \sqrt{25b^3} \), and \( \sqrt{49b^3} \) into simpler terms before doing further calculations.

Simplifying Radicals

Simplifying radicals means expressing the radical in its simplest form, which involves removing any perfect square factors under the square root. This process often follows the principle of prime factorization.
Here's a quick guide to simplify radicals:

• Identify and factor the number inside the root into its prime factors.
• Look for pairs of prime factors because \( \sqrt{a^2} = a \).
• Move each pair out of the root as a single number.

In our exercise:

• \( \sqrt{9b^3} = \sqrt{(3^2) (b^3)} = 3b^{3/2} \)
• \( \sqrt{25b^3} = \sqrt{(5^2) (b^3)} = 5b^{3/2} \)
• \( \sqrt{49b^3}= \sqrt{(7^2) (b^3)} = 7b^{3/2} \)

This simply means pulling out factors that appear twice under the square root symbol while reducing the power of the variable, \(b\), by half in each instance. Simplifying radicals makes it easier to manage the expressions as you proceed in solving or rearranging algebraic expressions.

Like Terms

Like terms are terms in an expression that have the same variables raised to the same power. They can be combined by simply adding or subtracting the coefficients. This concept is particularly important when you're simplifying expressions or solving equations.
In our exercise, after simplifying each radical, we were left with terms that looked like:

• \(3b^{3/2}\),
• \(5b^{3/2}\), and
• \(7b^{3/2}\).

These terms are considered "like terms" because they contain the same variable \(b\) raised to the same power \(3/2\).
Combining like terms involves:

• Add or subtract the coefficients of these terms while keeping the variable part unchanged.

So in our solution, we performed \((3 - 5 + 7)\) to find the final coefficient of \(b^{3/2}\), resulting in a simplified term \(5b^{3/2}\). Recognizing and correctly handling like terms is crucial in algebra for simplifying expressions and solving equations efficiently.